WRITTEN AREA COMPETITION ALGEBRA I ICTM REGIONAL 2018 ...

WRITTEN AREA COMPETITION ALGEBRA IICTM REGIONAL 2018 DIVISION A PAGE 1 OF 3

1. In a game, Dani has 4 red cards and 5 blue cards and Sammi has 2 red cards and 7 blue cards.Each red card is worth 6 points and each blue card is worth 3 points. Determine how manymore total points Dani has than Sammi.

2. Let 50 98 20k p+ + = where p is a prime integer. Determine the value of k .

3. The slope of a line is2

3and the intercepty − of the line is 2. An equation for this line can

be written in the form Ax By C+ = where A , B , and C are relatively prime integers with

0A > . Determine the value of C .

4. Determine the sum of all integer values of k such that2 7

5 7 8

k< < .

5. Define 2*a b a b= + and 3a b a b⊗ = + . Determine the value of ( )2* 3 4⊗ .

6. Cathy is twice as old as Don and3

4as old as Bob. Don’s age is 30% of Amanda’s age.

Bob’s age is k times Amanda’s age. Determine the value of k . Express your answer as aninteger or as a common or improper fraction.

7. One solution for x is 2 more than the other solution for x in 2 24 0x kx+ + = . Determine thelargest possible value of k .


8. Let ( )2

3 10x y+ = . Determine the numerical value of ( )2 26 36 54x xy y+ + .

9. A red, yellow and blue die are tossed and the numbers facing up on each die are recorded.Determine the probability that the yellow and blue die show the same number and the red dieshows a number that is greater than the common number on the yellow and blue die. Expressyour answer as a common fraction.

10. Determine the value(s) of x for which 19 27x+ = . Express your answer(s) as an integer or asa common or improper fraction.

11. A positive three-digit number has the property that, when the digits are reversed, it forms asecond three-digit number. When the second number is subtracted from the original number,the difference is 198. In the original number, the tens digit is 4 more than the units digit.Determine the sum of all such possible original numbers. (Note: 012 is not considered athree-digit number.)

12. An equilateral triangle has a side of length 8. A square has the same perimeter as theequilateral triangle. The length of one pair of opposite sides of the square is increased by 3and the length of the other pair of sides is left as is. Determine the numeric area of therectangle formed.

13. In a recent election for mayor, there were 3 candidates, Riemann, Euler and Gauss. The ratioof Gauss voters to Reimann voters is the same as the ratio of Reimann voters to Euler voters.There are 800 Gauss voters and 200 Euler voters and each voter voted for one and only oneof the three candidates. Determine the total number of voters.

14. Let ( )y f x= describes a linear function with ( )2 7f = and ( )4 5f − = − . Determine the

value of ( )2f − .


15. Given6 4 12 2

8 3 6 10

a c

k d

− + =

. The determinant of a 2 2× matrix is defined as:

x yxw yz

z w= − . Determine the value of the determinant

a c

d k.

16. Let 3 9x y+ = and 2 5 19x y+ = . Determine the ordered pair ( ),x y . Express your

answer as that ordered pair ( ),x y .

17. Determine the value(s) of x for which ( ) ( ) ( )2018 2018 2018 2018 2018 201820 18 20 18 18 20x− + − = −

18. A number is increased by 20% and this new value is decreased by a factor of2

3. The final

result is 120. Determine the original number.

19. The largest integral solution for x in ( )3 2 2 4x x x k− + < + is 18. Determine the smallest

possible integral value of k .

20. Two of the solutions for x in the equation 3 29 5 45x x x− = − are selected at random. These

two values are the solutions for y in the equation 2 0ay by c+ + = where a , b , and c are

integers with no common factors and 0a > . Determine the probability that 0b < . Expressyour answer as a common fraction.

2018 RA Name ANSWERS

Algebra I School(Use full school name – no abbreviations)

Correct X 2 pts. ea. =

Note: All answers must be written legibly in simplest form, according to the specificationsstated in the Contest Manual. Exact answers are to be given unless otherwisespecified in the question. No units of measurement are required.

("Points" or "pts."or "more" optional.)

1. 11.

2. 12.

("voters" optional.)

3. 13.

4. 14.

5. 15.

(Must be this (Must be thiscommon fraction.) ordered pair.)

6. 16.

(Must be thisanswer only.)

7. 17.

8. 18.

(Must be thiscommon fraction.)

9. 19.

(Must be this common (Must be thisfraction only.) common fraction.)

10. 20.

6

128

1

2

2

3

5

72 13

60 300

10 2−

4

5

2865

54

6− 1400

18 1−

35 120

( )4,9

WRITTEN AREA COMPETITION GEOMETRYICTM REGIONAL 2018 DIVISION A PAGE 1 OF 3

1. An isosceles triangle and a square both have the same non-zero base of length b and haveequal areas. The altitude to the base of the isosceles triangle is then length kb . Determinethe value of k .

2. Points A , B , C , D , and E line on a line in that order with AC CE≅ , 7AC = , 4DE =

and 3BC = . F is a point not on AE

such that FC AE⊥ and 158EDF∠ = ° . Determinethe degree measure of BFC∠ .

3. Quadrilateral ABCD is a trapezoid and has leg 9AD = , base 6CD =

and 2m D m B∠ = ∠ . Determine the length of base AB .

4. Point A is 6 units from one side of a rectangular billiards table (nopockets). Point B is on the opposite side of the table and 12 units measuredfrom the same side of the table. A ball is shot along a straight line frompoint A to a point P on the 80-unit side of the table and rebounds on astraight line to point B . Determine the exact length of the shortest pathfrom A to B . (Note: Figure not drawn to scale.)

5. In ADE∆ with BC DE , 5AB = , 3BD = , BC x= , and 6DE x= + .

Determine the length of segment DE .

6. K , M , R , S , and T are collinear such that T and M trisect KR , KM TM KT+ = , and

R bisects MS . 4MT x= + and 2 2 18KR x x= + − . Determine the length of TS .

7. Quadrilateral ABCD is a parallelogram. Q is the midpoint

of AB , R is the midpoint of QB , and S and T trisect

DC . Determine the ratio of the area of quadrilateral AESDto the area of RET∆ . Express your answer as an integer ora common or improper fraction.

B

C

D

A

612

A

B

P

C

A

D E

B

E

RQ B

CD

A

S T


8. EN is tangent to the circle at point N . 43ANE∠ = ° and 31ANU∠ = ° .Determine the degree measure of UAN∠ .

9. The sum of the measures of the interior angles of one convex polygon is 36times larger than the measure of one exterior angle of a second regular polygon with 4 moresides. Determine the number of diagonals in the first polygon.

10. BD is the altitude to the hypotenuse of right triangle ABC∆ . 10AB = and

25AC = . Determine the length of AD .

11. A square tile sits on top of a regular hexagonal tile so that the diagonal ofthe square exactly coincides with one of the longest diagonals of thehexagon. The area of the hexagon that shows from under the square tileis %k of the area of the hexagon. Determine the value of k . Expressyour answer as a decimal rounded to the nearest tenth and without usingthe % symbol.

12. Starting at noon, Jake rode his bike due east averaging 14 kilometers per hour. A half hourlater at 12:30, he turned due north and rode at 12 kph until 2:30 pm. At that point, he wasable to turn on a straight road that took him directly back to the original starting point,arriving there at 4:00 pm. Determine the average speed in kph for his entire ride.

13. Quadrilateral ABCD has vertices ( )2,6A , ( )3,2B , ( )5,5C and ( )4,11D . Determine the

numerical area of quadrilateral ABCD .

14. ABC∆ is a 45 45 90° − ° − ° triangle with right angle C . 2 2 2AB x= + and

2 10BC x= − . Determine the length of AC .

N

A

U

E

D

A

B C


15. A certain circle with equation 2 26 8 11 0x x y y+ + − − = can be graphed in the coordinate

plane with center ( ),h k and radius r . Determine the ordered triple ( ), ,h k r .

16. A circle is internally tangent in a right triangle with hypotenuse of length 82 and one leg oflength 80. Determine the radius of this circle.

17. In circle N , 3HK = 9EK = , and 5IK = . Determine the length of RK .

18. In ABC∆ , C∠ is a right angle, 160AC = and 120BC = . M lies on AB and K lies on

AC so that line KM

is the perpendicular bisector of AB . Determine the length of KM .

19. Quadrilateral ABCD is a rectangle with 4AB = and 2BC = .Segments from B to the midpoints P and Q of the two other sides

are drawn to form quadrilateral BPDQ . Determine the area of

quadrilateral BPDQ .

20. AB , CD , and EF are parallel chords on the same side of the center of a

circle. The distance between AB and CD is equal to the distance

between CD and EF . 20AB = , 16CD = , and 8EF = . The radius of

this circle is then expressed ask w

rp

= in reduced radical form.

Determine the sum ( )k w p+ + .

Q

P

D

B

C

A

D

F

A B

C

E

K

N

E

I

H R


Geometry School (Use full school name – no abbreviations)

Correct X 2 pts. ea. = Note: All answers must be written legibly in simplest form, according to the specifications

stated in the Contest Manual. Exact answers are to be given unless otherwise specified in the question. No units of measurement are required.

(Must be this decimal.) 1. 11. ("degrees" or (“kilometers per hour” " " optional.) or “kph” optional. 2. 12. 3. 13. 4. 14. (Must be this ordered triple.) 5. 15. 6. 16. (Must be this reduced improper fraction.) 7. 17. ("degrees" or " " optional.) 8. 18. (“diagonals” optional.) 9. 19. 10. 20.

2

4 29

20 4

106 75

22

9 15

24.4

68 14

15 14

30 8

82 (also accept 2 1069

16

14

3,4,6

WRITTEN AREA COMPETITION ALGEBRA IIICTM REGIONAL 2018 DIVISION A PAGE 1 OF 3

1. ( ) ( )7 3 22 8k k+ = . Determine the value of k .

2. Line AB

contains points ( )1,4A − and ( )5,B k and is perpendicular to the line2

53

y x= + .

Determine the value of k .

3. The 7th term of an arithmetic sequence is 4− . The 77th term is 214− . Determine the sum ofthe first 777 terms.

4. Determine the quotient of the positive geometric mean of 6 and 12 divided by the arithmeticmean of 6 and 12.

5. ( )3 33 log 3 2 5 2logk k+ + = + . Determine the sum of all possible value(s) of k . Express

your answer as an integer or as a common or improper fraction.

6. The product of 3 consecutive odd integers is equal to 3 times the square of the middleinteger. Determine the sum of these three consecutive odd integers.

7. The distance between the two lines 2 3 1 0x y− + = and 3 2 5y x− = , when in simplest radical

form, is written ask w

p. Determine the sum ( )k w p+ + .

8. Let ( ) 2 3f x x= + . ( )1 4f k− = . Determine the value of k .


9. x and y are positive real numbers with 4 5 20x y+ = . Determine the maximum possible

value of the product ( )xy .

10. Determine the sum of all possible solution(s) for 2 7 1x x x− − = + .

11. x varies inversely as y and jointly as z and the cube of w . The constant of variation is 64.

Determine the value of x when z y= , 2w x= , and 4y x= . Express your answer as an

integer or a common or improper fraction.

12. When

2

4 1

23

x xy

x

−+= is written as a single simplified rational expression, it is in the form

2ax bx cy

dx e

+ +=

+. Determine the sum ( )a b c d e+ + + + .

13. Determine the positive value of k so that 23 27 0x kx+ + = has only one (double) root.

14. A number written in a particular number base is written as 264. When written in a differentnumber base, the same number is 1030. These two bases differ by 5. Determine the numberin base 10. Express you answer as that number in base 10.


15. Three squares are constructed along the x-axis beginning at theorigin and share a vertex with the neighbor square, and the side

lengths form a geometric sequence. ( )0,8A is a vertex on the

first square and ( ),50B k is a vertex on the third square as

shown. Determine the value of k .

16. Six masons working at the same rate can tile 540 square feet of floor in 4 hours. Working atthe same rate, determine how long, in hours, it would take 9 masons to tile 1620 square feetof floor. Express your answer as an integer or as a common or improper fraction reduced tolowest terms.

17. The difference of the roots of 2 306 0x kx+ + = is one and 0k > . Determine the value of k .

18. ( )( ) ( )( ) ( )( )2 3 5 5 2 3 3 5 2log log log log log log log log log 0x y z= = = . Determine the sum

( )x y z+ + .

19. When graphed, the parabolic equation 2y ax bx c= + + has zeros 1− and 3 and passes

through the point ( )0, 9− . Determine the ordered triple ( ), ,a b c . Express your answer as

this ordered triple.

20. The solutions to the system

5

2

4

0

x y

x y

x

y

+ ≥ ≥

≤ ≥

are graphed and form a polygonal region in the

coordinate plane. Determine the area of this region containing the solutions to the system.Express your answer as an integer or a common or improper fraction.

x-axis

y-axis

B(k,50)

A(0,8)


Algebra II School (Use full school name – no abbreviations)

Correct X 2 pts. ea. = Note: All answers must be written legibly in simplest form, according to the specifications

stated in the Contest Manual. Exact answers are to be given unless otherwise specified in the question. No units of measurement are required.

(Must be this reduced common fraction.) 1. 11. 2. 12. (Comma usage optional.) 3. 13. 4. 14. (Must be this reduced common fraction.) 5. 15. (“hours” or “hr.s” optional.) 6. 16. 7. 17. 8. 18. (Must be this ordered pair.) 9. 19. (Must be this reduced common fraction.) 10. 20.

3

2 1

3

5 3, 6, 9

11 166

30 35

1

4

5 20

3 8

893550 18

22

3 OR

2 2

3

2

3 78

364 OR 364ten OR 10364

WRITTEN AREA COMPETITION PRECALCULUSICTM REGIONAL 2018 DIVISION A PAGE 1 OF 3

1. An infinite geometric series with first term 1 4g = has a sum of 10. Determine the common

ratio for this series. Express your answer as an integer or as a common or improper fraction.

2. Let ( )fcn represent one of the six elements from set { }sin ,cos , tan ,cot ,sec ,cscT x x x x x x= .

Then the expression( ) ( )

( )2

2

sec 1 sec 1

sin

x xfcn

x

+ − = . Determine which element of set T is

the appropriate substitution for ( )fcn . Express your answer as that element.

3. ( ),k w is the solution to the matrix equation3 4 3

2 3 4

x

y

=

. Determine the value of the

sum ( )k w+ .

4. The area between the curves 6cosr = θ and 2cosr = θ is kπ . Determine the value of k .

5. Let ( )6,4a =

, ( )14,9b =

, ( )36,43c =

, and ( ),d x y=

be vectors such that a b c d+ = −

.

Determine the components of d

. Express your answer as the ordered pair ( ),x y .

6. Let k be a positive integer such that ( )( )6 9 362 3 k= . Determine the value of k .

7. Two lines 1

13

4y x= − and 2

15

2y x= + intersect. Determine the degree measure of the

obtuse angle formed by this intersection. Express your answer as a decimal rounded to thenearest tenth of a degree.


8. 1i = − . Determine the value of 11 60i− .

9. In interval notation, [ ],k w is the domain for x for the real-valued conic section

( )2 23

116 25

x y−+ = . Determine the value of the sum ( )k w+ .

10. Vectors 2 3v i j= +

and 6u i k j= − +

. Determine the value of k so that v u⊥

.

11. Box A contains 3 red marbles and 6 blue marbles. Box B contains 2 red marbles and 1 bluemarble. Box C contains 2 red marbles and 4 blue marbles. Two of the boxes are randomlychosen. Then one marble is chosen from each of those two boxes. Determine the probabilitythat these two marbles drawn are the same color. Express your answer as a common fraction.

12. Consider the graph of the function ( ) ( )3sin 2 5f x x π= + − . Determine the amplitude of the

graph of ( )f x .

13. A bird sitting on a branch 25 feet above level ground sees a worm on the ground at a 52°angle of depression. Determine the straight-line distance from this bird to the worm.Express your answer as a decimal rounded to the nearest tenth of a foot.

14. The sum of all real values of x , 0 2x π≤ ≤ , that satisfy 212sin sin 1x x= + is kπ .Determine the value of k . Express your answer as an integer or as a common or improperfraction.


15. The radian measure of the angle coterminal with204

5π that falls between 0 and 2π is kπ .

Determine the value of k . Express your answer as an integer or as a common or improperfraction.

16. ( ) ( )sinf x kx= and ( ) ( )tang x wx= where ( ) 6k w+ = . Then there exist ordered pairs

( ),k w such that ( )f x and ( )g x have the same period. Determine the sum of all possible

values of these k .

17. Points A and B lie on circle O . The radius of circle O is 12. 10AB = . Determine the

length of AB . Express your answer as a decimal rounded to the nearest tenth.

18. Let k and w be positive numbers such that their arithmetic mean and geometric mean have a

sum of 18. Determine the sum ( )k w+ .

19.( )4

lim5

x

x

x ek

x

−

→∞

+= . Determine the value of k . Express your answer as an integer or a

common or improper fraction.

20. A plane flying at an average speed of 420 mph at an angle of 40° west of due northencounters a wind pushing 20 mph at an angle of 20° west of due south. The plane will thenbe traveling at a new angle west of due north. Determine the degree measure of that angle.Express your answer as a decimal rounded to the nearest hundredth.


Pre-Calculus School(Use full school name – no abbreviations)



(Must be this reduced (Must be this reducedcommon fraction.) common fraction.)

1. 11.

(Accept “sec”or “secant”.)

2. 12.

(Must be this decimal,“ft.” or “feet” optional.)

3. 13.

4. 14.

(Must be this (Must be this reducedordered pair.) common fraction.)

5. 15.

6. 16.

"degrees" or

" "° optional.) (Must be this decimal.)

7. 17.

8. 18.

(Must be this reducedcommon fraction.)

9. 19.

("degrees" or

" "° optional.)

10. 20.

3

5

4 42.42

64

5

61 6

115.3 10.3

5184 16

( )16,30

13

27

sec x 3

1− 31.7

4

5

8 4

FROSH-SOPH EIGHT PERSON TEAM COMPETITION ICTM REGIONAL 2018 DIVISION A PAGE 1 OF 3

NO CALCULATORS

NO CALCULATORS NO CALCULATORS NO CALCULATORS

1. Determine the solution ,x y for the system 3 8 28

2 6

x y

x y

. Express your answer as

that ordered pair ,x y .

2. Consider the number set 71,77,81,83,87S . One element of S is selected at random.

Determine the probability that element is a prime number. Express your answer as an integer or as a common fraction reduced to lowest terms.

3. A standard square graph grid is painted on a large, level playground. Start at the origin.

Face the positive direction of the x-axis. Move 1 unit in the positive x direction to point

1,0 . Rotate your body 90 counterclockwise and move 2 units up to 1,2 . Rotate your

body 90 counterclockwise again and move 3 units left to the next point 2,2 . Continue

this process of rotation and moving one more unit each for moves of 4, 5, 6, 7, 8, 9, and 10 units, ending at point P . Determine the direct distance from P to the origin.

4. Determine the exact value of 3

2 221 2 8 5 3 . Express your answer as an integer

or as a common or improper fraction. 5. The arithmetic mean of set 2,4,7,A k is 8 and the median of this set is 5.5 . Determine

the range of set A . Express your answer as an integer or as an exact decimal.

6. When 0x , 3 9 27

3 9 27

x x x xk w p x

x x x x in simplified radical form.

Determine the sum k w p .


NO CALCULATORS


A B

C

F

D

G

E

7. In rectangle ABCD , AB EF , AE AD and AB EF

AD FC . The exact

ratio of k w pAB

AD q

where k , w , p , and q are relatively prime

integers with 0q . Determine the sum k w p q .

8. Twice A is directly proportional to the square of B and inversely proportional to the square

root of the product of four and C . If 2A and 6B when 9C , then determine the value of C when 9A and 3B . Express your answer as an integer or as a common or improper fraction.

9. Determine the remainder when 20187 is divided by 5.

10. 3 2 3 23 1 3 1x x kx wx px q . Determine the value of k w p q .

11. The locus of points ,x y in the coordinate plane that fit the criterion that the positive

difference of the distances from points 7,3 and 1,3 is always 4 can be written as 2 2 0Ax Bxy Cy Dx Ey F where A , B , C , D , E , and F are relatively prime

integers and 0A . Determine the sum A B C D E F .

12. In the diagram (not drawn to scale), AB CD , 35BGF , and 125GFE . Determine the degree measure of FED .

13. Let x be an odd integer less than one and let y be an even integer less than zero. Determine

the largest possible value of 5 2x y . Express your answer as an integer or as a common

or improper fraction.

BA

D

E

F C


NO CALCULATORS


14. Determine the sum of all possible value(s) for k so that the system

3 6

4 4

kx y

x y k

x y

is

consistent. Express your answer as an integer or as a common or improper fraction. 15. Consider the first ten positive integers. When one of these integers is subtracted from the

sum of the other nine integers, the result is 41. Determine the integer that was subtracted. Report as your answer that integer that was subtracted.

16. The polynomial 26 5x x k is divided by 3 4x and leaves a remainder of 10 .

Determine the value of k . 17. Determine the number of composite positive integers less than 50 that have irrational square

roots.

18. Determine the value of x so that 279 9 9 27x x x . 19. A pair of standard 6-sided dice are rolled. Determine the probability of rolling either at least

one 6 or a sum of 6 shown on the face-up dice. Express your answer as an integer or as a common fraction reduced to lowest terms.

20. A cook at a summer camp knows how to prepare two different dishes for dinner. He

prepares one of those dishes for dinner each day. The camp only allows him to cook any one dish no more than two days in a row. Determine the number of different 10-day dinner menus he could prepare.

2018 RA School ANSWERS

Fr/So 8 Person (Use full school name – no abbreviations)



(Must be thisordered pair.)

1. 11.

(Must be this reduced ("degrees" or

common fraction.) " "° optional.)

2. 12.


3. 13.

(Must be this reduced (Must be this reducedcommon fraction.) common fraction.)

4. 14.

5. 15.

6. 16.

(“integers” optional.)

7. 17.


8. 18.


9. 19.

(“menus” optional.)

10. 20.

( )4,2

2

5

611

27

17

11

9

1

36

4

48 178

4

9

40

27

14−

7

89

90

9

20

1

4− OR

1

4

−

JUNIOR-SENIOR EIGHT PERSON TEAM COMPETITION ICTM REGIONAL 2018 DIVISION A PAGE 1 OF 3

NO CALCULATORS


1. Let 2log 5 x and 2log 9 y . Then 2log 243000 ax by c . Determine the sum

a b c . Express your answer as a decimal.

2. Determine the positive number whose square exceeds its square root by 252.

3. 7

17

17

11

k w p

q

where k , w , and q are relatively prime integers with 0q .

Determine the sum k w p q .

4. Determine the sum of the solution(s) for x when 2 2 22 4 6

2 3 5

x x x x x . Express your

answer as an integer or as a common or improper fraction.

5. Given that 2 2 2x y x y , 1 1x , and 2 2y , determine the largest possible

value of 4 xy . 6. The polynomial 3 215 80f x x x kx has integral roots that form an arithmetic

sequence. Determine the value of k . 7. For the real number 1, arrange the expressions sin1, cos1 , and tan1 in order from smallest

to largest. Report for your answer the expression that is the largest of these three. 8. The equation y xx y has the integer solution x m and y p , where m p . Determine

the value of 2pm .


NO CALCULATORS


9. 1a , 2a , 3a , 4 ,a is an arithmetic sequence and 1g , 2g , 3g , 4 ,g is a geometric sequence.

Then 1 1a g , 2 2a g , 3 3 ,a g is called a hypergeometric sequence. Determine the sum of the

hypergeometric infinite series 2 3 4

1 1 1 13 5 7 9

2 2 2 2


an integer or as a common or improper fraction. 10. Let w be a solution for x to the equation tan cot 4x x . Then sin 2k w .

Determine the value of k . Express your answer as an integer or as a common or improper fraction.

11. The large circle O contains the three smaller circles P , Q , and R so

that the three smaller circles are internally tangent to the larger circle and at the same time externally tangent to each other as shown. Q and R lie on a diameter. The radius of Q is 2, and the radius of R is 1. Determine the radius of circle P . Express your answer as an integer or as a common or improper fraction.

12. The reciprocal of one more than the reciprocal of a number is 2. Determine the value of this

number.

13. Express 2

2 3 in simplified and reduced radical form.

14. Determine the number of odd integers between 1 and 10000, inclusive, that can be formed

using the 10 digits 0-9 and using each of the chosen digits at most once in each number. (Note: By convention, leading zeros do not make a new number. The number 02018 is not considered different from 2018.)

15. Determine the value of x so that 12! 8!5! 450x x . Express your answer as an integer

or as a common or improper fraction.

O

P

Q R


NO CALCULATORS


16. Let the “Sudoku Sum” of a square matrix n nM be defined by 1 2 nSS M m m m

where 1m is chosen at random from the entries of M after which the rest of 1m ’s row and

column entries are deleted. Next, 2m is chosen at random from the remaining entries after

which what is left of 2m ’s row and column entries are deleted. The process continues,

choosing im from remaining entries and then deleting 'im s remaining row and column

entries, until the final entry is chosen. Let

1 100

1 100

1 100

1 100

A

and

1 100

101 200

9,901 10,000

B

. Determine the value of SS A SS B .

17. 4

1

1lim

1x

xk

x

. Determine the value of k . Express your answer as an integer or as a


18. Determine the sum of all integers such that the expression 4

2

81

12 36

x

x x

is negative valued.

19. 1Tan x represents the inverse tangent function. Then 1Tan 1 k . Determine the value

of k . Express your answer as an integer or as a common or improper fraction.

20. 2

3

3

2n

kn n




Jr/Sr 8 Person (Use full school name – no abbreviations)



(Must be this reduced(Must be this decimal.) common fraction.)

1. 11.

2. 12.(Parens useoptional.)

3. 13.

(Must be this reduced (“odd integers” orimproper fraction.) “integers” optional.)

4. 14.

(Must be this reducedimproper fraction.)

5. 15.

(Comma use optional.)

6. 16.

(Must be this expression,caps and/or parens optional.)

7. 17.

8. 18.


9. 19.

(Must be this reduced (Must be this reducedcommon fraction.) improper fraction.)

10. 20.

8.5

16

33

36

19

1

66

tan1

256

5

1

211

6

1

4− OR

1

4

−

0 OR zero

4

495000

9

2

6

7

2−

6 2− OR 2 6− +

2605

CALCULATING TEAM COMPETITION ICTM REGIONAL 2018 DIVISION A PAGE 1 of 3 Round answers to four significant digits and write in standard notation unless otherwise specified in the question. Except where noted, angles are in radians. No units of measurement are required.

1. The ordered pair ,x y is a solution to the system 0.147 0.235 5.0192

1.38852 7.6 5.793482

x y

x y

.

Determine the value of y in that solution. 2. It is known that 17log ( )k is a positive integer greater than 2.4693 . Determine the smallest

possible value of k . 3. In a certain Algebra class, exactly 10% of the students are 15 years old, exactly 25% of the

students are 14 years old, and exactly 65% of the students are 13 years old. The positive difference between the class arithmetic mean age and the class median age is an exact decimal. Determine that decimal.

4. On a specific day, the conversion rate between the US Dollar (USD) and the South African

Rand (RAND) was 1USD 12.87 RAND . At a particular location, a bank paid 12.75 RAND per USD, the difference being the cost of making the exchange. An American tourist exchanged $500 USD for RANDs. Determine the amount, in USD, the tourist paid to make the exchange. Express your answer in standard dollar and cent notation, rounded to the nearest cent.

5. The first two Fibonacci numbers are 1 and 1, and each subsequent Fibonacci number is the

sum of the previous two. The thn Fibonacci number can be approximated by

1 1 5

25

n

F n

. Determine the positive difference between the 21st Fibonacci

number and the approximation for that number. Express your answer in scientific notation. 6. $5000 is invested at an annual percentage rate (APR) of %r compounded monthly.

Determine the value of r that would make the effective annual yield (APY) 4.6% . Express your answer as the value for r only. Do not include the percent symbol.

CALCULATING TEAM COMPETITION ICTM REGIONAL 2018 DIVISION A PAGE 2 of 3

7. The diagonals of a quadrilateral have lengths of 37 and 54. They meet at point O such that one diagonal is divided into segments of 10 and 27, while the second diagonal is divided into segments of 17 and 37. Determine the numeric maximum possible area for this quadrilateral. Express your answer as an integer or as a common or improper fraction.

8. 1i . Let 2 0.2 0.1f x x i and g x f f f f f f x . Determine

0.1 0.3g i . Express your answer in standard a bi form with arguments a and b

rounded to the nearest hundredth.

9. The values not in the range of 2 2x x

f xx

can be expressed as the interval ,k w .

Determine that interval. Express your answer in interval notation. 10. A council of 5 men and 6 women randomly select a committee of three members. Determine

the probability that the committee will contain at least one member from each gender. Express your answer as an integer or as a common fraction.

11. The diagram, not drawn to scale, shows a square flower garden with

a circular sidewalk internally tangent to the square boundary. Sidewalks also cut through the garden, symmetric and perpendicular to each other creating four equal smaller garden plots. All sidewalks are 1-yard wide. The diameter of the outside of the circular sidewalk is 9 yards. The numeric area covered by the sidewalk (the unshaded area) is k square yards. Determine the value of k .

12. Let 1 1a and for 2n , 11 sinn na a where na is measured in radians. Determine the

2018th term of this sequence.

13. Determine the value of x such that 5

2 2 5 28 4 202

x x x


an integer or as a common or improper fraction.

CALCULATING TEAM COMPETITION ICTM REGIONAL 2018 DIVISION A PAGE 3 of 3

14. Determine the value of 34

2

1 1

p

k p

k p

. Express your answer as an integer or as a common or

improper fraction. 15. Determine the sum of the zeros for the polynomial 4 1 3 23 2 7p x x e x x x when

solved over the set of Complex Numbers. 16. Determine the amplitude of the graph of 2.835sin 8.241cosy .

17. The fifth power of an integer is 6 , , 7xx xxx xx where the x represents missing but not

necessarily the same digit. Determine that integer.

18. The continued exponential expression

xx

xe

. Determine the value of x . 19. A multiplier doctrine in economics suggests continued benefits from an income tax rebate.

That is, the recipient spends a percentage of the rebate in the community, the recipients of that spending again spend the same percentage of what they received back into the community, and so forth. Suppose a person receives a $500 tax rebate and spends 80% back into the community. Determine the total dollar value to the community.

20. In ABC , 8AC , 6BC , and 48BAC . Determine the sum of all possible numeric

area(s) for distinct triangle(s) ABC .


Calculator Team (Use full school name – no abbreviations)


Note: All answers must be written legibly. Round answers to four significant digits andwrite in standard notation unless otherwise specified in the question. Exceptwhere noted, angles are in radians. No units of measurement are required.

1. 11.

2. 12.(Must be this exact

decimal, “years” (Must be this reducedoptional.) improper fraction.)

3. 13.

(“$”, “USD” (Must be thisor “dollars” optional.) integer.)

4. 14.(Must be in scientificnotation, parens inexponent optional.)

5. 15.

(Must be this decimalwith no “%” sign.)

6. 16.

(Must be this (Must be thisinteger.) integer.)

7. 17.

8. 18.

(Must be this ordered (“dollars” or “$” optional,pair interval.) accept 2000.00)

9. 19.


10. 20.

7.735−

4913

0.45 OR .45

4.66

( )51.827 10

−×

4.506

999

0.22 0.18i+ OR .22 .18i+

( )1.828,3.828−

9

11 31.82

2000

1.038

57

8.715

0.9061− OR .9061−

140

19

18− OR

19

18

−

1.935

12.45

ICTM Math Contest

Freshman – Sophomore

2 Person Team

Division A

FROSH-SOPH 2 PERSON COMPETITION LARGE PRINT QUESTION 1 ICTM 2018 REGIONAL DIVISION A NO CALCULATORS ALLOWED

1. Let 22 5f x x .

Determine the value of 6 4f f .


2. Let

2

2 3 1 2 5x x x x

ax bx c

.

Determine the ordered triple , ,a b c . Express your answer as that ordered triple.


3. Let x and y be the

lengths of legs of a right triangle where 3 10 7

5 2

x x and

4 30

8 5

y y .

Determine the length of the hypotenuse of this triangle.


4 y + 1( )7 y - 2( )

5 x - 6( )

3x - 12H

TA

M

4. Determine the perimeter

of parallelogram MATH with side lengths marked as shown in the diagram.


5. Point A has coordinates 4,a and lies on the line

2 6y x . Point B has coordinates ,6b and lies on the line 2 8y x . Determine the length of AB.

FROSH-SOPH 2 PERSON COMPETITION LARGE PRINT QUESTION 6 ICTM 2018 REGIONAL DIVISION A CALCULATORS ALLOWED

6. A circle with radius r has

an area of 20. A circle with radius 2r has an area of K . A circle with

radius 1

2r has an area of

W . Determine the difference K W .


7. The arithmetic mean of

the numbers 7, 8, 11, 13, 21 is K . The positive geometric mean of the numbers 3 and 12 is W . Determine the arithmetic mean of K and W .


8. A n n square is divided

into 1 1 squares. In this square, there are a total of 1,015 squares of any side length. Determine the value of n.


9. A pentagon has exterior angles in

the ratio 2 : 2 :3:3:5. Let K represent the degree measure of the smallest interior angle of this pentagon.

A triangle has interior angles in the ratio 2 :3: 7. Let W represent the degree measure of the smallest interior angle of this triangle.

A triangle has interior angles with degree measures of Kand W . Determine the degree measure of the third angle of this triangle.


10. Two sides of a non-right,

scalene triangle have lengths of 5 and 13. Let N represent the number of possible integral lengths of the third side of the triangle. Let W represent the number of integral values of x which are solutions to 2 3 1 10x . Determine

the sum N W .

FROSH-SOPH 2 PERSON COMPETITION EXTRA LARGE PRINT QUESTION 11 ICTM 2018 REGIONAL DIVISION A CALCULATORS ALLOWED

11. Determine the number

of pathways starting from A and continuing to C that pass through point B, moving only to the right or down along the grid lines.

C

B

A

FROSH-SOPH 2 PERSON COMPETITION EXTRA LARGE PRINT QUESTION 12 ICTM 2018 REGIONAL DIVISION A CALCULATORS ALLOWED

12. The Let k be the sum of the distinct members of the following set that are rational:

2 2 3

{5, , 3, 121,

( 17) , (15) , 4}

Let the areas of two similar triangles be in the ratio of 1: 4. Let w be the length of the longest side of the smaller triangle if the length of the corresponding side in the larger triangle is 24. Determine the value of k w .

FROSH-SOPH 2 PERSON COMPETITIONICTM 2018 REGIONAL DIVISION A PAGE 1 OF 2

4 y + 1( )7 y - 2( )

5 x - 6( )

3x - 12H

TA

M

1. Let ( ) 22 5f x x= − . Determine the value of ( ) ( )6 4f f− .

2. Let ( )( ) ( )( ) 22 3 1 2 5x x x x ax bx c+ − + + + = + + . Determine the ordered triple ( ), ,a b c .

Express your answer as that ordered triple.

3. Let x and y be the lengths of legs of a right triangle where3 10 7

5 2

x x+ += and

4 30

8 5

y y− −= . Determine the length of the

hypotenuse of this triangle.

4. Determine the perimeter of parallelogramMATH with side lengths marked as shown inthe diagram.

5. Point A has coordinates ( )4,a and lies on the line 2 6y x= − . Point B has coordinates

( ),6b and lies on the line 2 8y x= − + . Determine the length of AB .

6. A circle with radius r has an area of 20. A circle with radius 2r has an area of K . A

circle with radius1

2r has an area of W . Determine the difference ( )K W− .

7. The arithmetic mean of the numbers 7, 8, 11, 13, 21 is K . The positive geometric meanof the numbers 3 and 12 is W . Determine the arithmetic mean of K and W .

8. A n n× square is divided into 1 1× squares. In this square, there are a total of 1,015

squares of any side length. Determine the value of n .

9. A pentagon has exterior angles in the ratio 2 : 2 : 3 :3 : 5 . Let K represent the degreemeasure of the smallest interior angle of this pentagon. A triangle has interior angles inthe ratio 2 : 3 : 7 . Let W represent the degree measure of the smallest interior angle ofthis triangle. A triangle has interior angles with degree measures of K and W .Determine the degree measure of the third angle of this triangle.

10. Two sides of a non-right, scalene triangle have lengths of 5 and 13. Let N represent thenumber of possible integral lengths of the third side of the triangle. Let W represent the

number of integral values of x which are solutions to 2 3 1 10x − + < . Determine the

sum ( )N W+ .

FROSH-SOPH 2 PERSON COMPETITION EXTRA QUESTIONS 11-15ICTM 2018 REGIONAL DIVISION A

11. Determine the number ofpathways starting from A andcontinuing to C and pass throughpoint B moving only to the rightor down along the grid lines.

12. The Let k be the sum of the distinct members of the following set that are

rational: 2 2 3{5, , 3, 121, ( 17) , (15) , 4}π − − . Let the areas of two similar

triangles be in the ratio of 1: 4 . Let w be the length of the longest side of thesmaller triangle if the length of the corresponding side in the larger triangle is 24.

Determine the value of ( )k w+ .

C

B

A


Fr/So 2 Person Team (Use full school name – no abbreviations)

Total Score (see below*) =


Answer Score(to be filled in by proctor)

1.(Must be this ordered triple.)

2.

3.

4.

5.

6.

7.

8.("degrees" or " "° optional.)

9.

10.

TOTAL SCORE:(*enter in box above)

Extra Questions:

(“paths” or “pathways” optional.)11.

12.

13.

14.

15.

* Scoring rules:

Correct in 1st minute – 6 points

Correct in 2nd minute – 4 points

Correct in 3rd minute – 3 points

PLUS: 2-point bonus for being firstIn round with correct answer

40

( )3,8,7

25

86

5

75

9

14

90

15

1680

30

(Intentionally blank.)



NOTE: Questions 1-5 onlyare NO CALCULATOR

ICTM Math Contest

Junior – Senior

2 Person Team

Division A

JUNIOR-SENIOR 2 PERSON COMPETITION LARGE PRINT QUESTION 1 ICTM 2018 REGIONAL DIVISION A NO CALCULATORS ALLOWED

1. Let

3 1 5 134

4 2 9 8

c d

a b

and 42log 3x . Determine the sum x a b c d .


2. Determine the value of

3 10.4 0.716 16


3. Let 2 2 3f x x x and 2 3 4g x x x . Determine the value of the sum

2 2f g g f .


4. ABC has a right angle at C, 30AB and

24AC . Determine the ratio of cosB to cos A. Express your answer in the form :x y where x and y are integers with no common factors.


5. Let

2

2

2 7 15

3 4

x xf x

x x

.

Then the domain of f x is ,x a x b and

limx

f x c

. Determine

the value a b c .

JUNIOR-SENIOR 2 PERSON COMPETITION LARGE PRINT QUESTION 6 ICTM 2018 REGIONAL DIVISION A CALCULATORS ALLOWED

6. An infinite geometric series has a common ratio

of 3

4 and a first term that is

an odd integer between 6 and 20. Determine the average of all possible sums of these series.


7. Point , 6P x is located on the axis of symmetry of the graph of

2 8 12y x x .

Point Q 2, y is located on the circle

2 21 2 17x y .

Determine the largest possible length of PQ.


8. A sequence is defined by 2

4 1 3 2na n n . Determine the absolute value of the difference between 20a and 18a .


9. One of the solutions for x in 5 313 36 0x x x is selected at random. Determine the probability that the solution selected is also a solution for x in

2 6 0x x . Express your answer as an integer or as a common fraction.


10. When 5

2x y is expanded, one of the terms is 3 cax y .

Let d be the length of the diameter in the circle

2 2 4 10 7 0x y x y .

Determine the sum a c d .

JUNIOR-SENIOR 2 PERSON COMPETITION EXTRA LARGE PRINT QUESTION 11 ICTM 2018 REGIONAL DIVISION A CALCULATORS ALLOWED

11. Let

403

0

3

2k

kk

kf x x

.

Determine the remainder when f x is divided by

2x .

JUNIOR-SENIOR 2 PERSON COMPETITION EXTRA LARGE PRINT QUESTION 12 ICTM 2018 REGIONAL DIVISION A CALCULATORS ALLOWED

12. Determine the minimum number of distinct points in three-dimensional space that could determine 84 distinct planes.

JUNIOR-SENIOR 2 PERSON COMPETITIONICTM 2018 REGIONAL DIVISION A PAGE 1 OF 2

1. Let3 1 5 13

44 2 9 8

c d

a b

+ = −

and 42log 3x = . Determine the sum ( )x a b c d+ + + + .

2. Determine the value of ( )( ) ( )( )3 1

0.4 0.716 16−

.

3. Let ( ) 2 2 3f x x x= − − and ( ) 2 3 4g x x x= + − . Determine the value of the sum

( )( ) ( )( )2 2f g g f + .

4. ABC∆ has a right angle at C , 30AB = and 24AC = . Determine the ratio of cos B tocos A . Express your answer in the form :x y where x and y are integers with no common

factors.

5. Let ( )2

2

2 7 15

3 4

x xf x

x x

− −=

+ −. Then the domain of ( )f x is ,x a x b≠ ≠ and ( )lim

xf x c

→∞= .

Determine the value ( )a b c+ − .

6. An infinite geometric series has a common ratio of3

4and a first term that is an odd integer

between 6 and 20. Determine the average of all possible sums of these series.

7. Point ( ), 6P x is located on the axis of symmetry of the graph of 2 8 12y x x= − + . Point Q

( )2, y is located on the circle ( ) ( )2 2

1 2 17x y− + + = . Determine the largest possible length

of PQ .

8. A sequence is defined by ( )2

4 1 3 2na n n= − + − . Determine the absolute value of the

difference between 20a and 18a .

9. One of the solutions for x in 5 313 36 0x x x− + = is selected at random. Determine the

probability that the solution selected is also a solution for x in 2 6 0x x− − ≤ . Express youranswer as an integer or as a common fraction.

10. When ( )5

2x y+ is expanded, one of the terms is 3 cax y . Let d be the length of the

diameter in the circle 2 2 4 10 7 0x y x y+ − + − = . Determine the sum ( )a c d+ + .

JUNIOR-SENIOR 2 PERSON COMPETITION EXTRA QUESTIONS 11-15ICTM 2018 REGIONAL DIVISION A PAGE 2 OF 2

11. Let ( ) ( )( )

403

0

3

2

k

kk

kf x x

=

+ =

∑ . Determine the remainder when ( )f x is divided by ( )2x − .

12. Determine the minimum number of distinct points in three-dimensional space that coulddetermine 84 distinct planes.


Jr/Sr 2 Person Team (Use full school name – no abbreviations)




1.

2.

3.(Must be this ratio in this form.)

4.

5.

6.(Must be this exact answer.)

7.

8.(Must be this reduced common fraction.)

9.

10.


Extra Questions:

(Comma usage optional.)11.

(“points” optional.)12.

13.

14.

15.

* Scoring rules:





32

4

17

3: 4

1−

52

2 37

2944

5

94

82618

9





ORAL COMPETITION ICTM REGIONAL 2018 DIVISION A 1. Given ABC with C 90 , B 27 and BC 16 . Find the length of the hypotenuse of

this triangle. Give your answer rounded to the nearest hundredth. 2. A 15-foot long ladder is leaning against a building. The top of the ladder is 8 feet off the ground.

If the top of the ladder slides 3 feet down the building, what is number of degrees in the change in the angle of elevation at the base of the ladder? Give your answer in degrees/minutes/seconds form, rounded to the nearest second.

3. The Bermuda Triangle has sides whose lengths are approximately 850 miles, 925 miles, and

1300 miles. Find the degree measure of the largest angle of the Bermuda Triangle. Round your answer to the nearest tenth of a degree.

4. Harry is playing the 17th hole on a golf course. The hole’s length from tee (T) to the pin (P) is

182 yards, and includes hitting over a large lake. Harry can’t hit the ball that far, and decides to play around the lake. On his first shot, he hit the ball 170 yards on a bearing of N 60 E to point B. On his second shot, he hit the ball on a bearing of N 157 E and the ball went in the hole at the pin (P). Find the measure of BPT . Round your answer to the nearest degree.

170

182

B

T P

24

40°C B

A

θ10

6

ORAL COMPETITION ICTM REGIONAL 2018 DIVISION A EXTEMPORANEOUS QUESTIONS Give this sheet to the students at the beginning of the extemporaneous question period. STUDENTS: You will have a maximum of 3 minutes TOTAL to solve and present your solution to these problems. Either or both the presenter and the oral assistant may present the solutions. 1. Using the given diagram,

Find the value of sin , cos and tan 2. Given triangle ABC as shown, set up equations for two

different ways you could find the length of BC . You do NOT need to do the calculations to find the length.

3. In each scenario for ABC listed below, comment on whether it is possible to draw two different

triangles with the given information. a. BC 7.0, A 110 , C 33 b. BC 23, AB 11, A 122 c. BC 8.1, AC 8.3, A 42

I.C.T.M. 2018 Regional Algebra I – Divisions 1A, 2A

I.C.T.M. 2018 Regional Algebra I – Divisions 1A, 2A 1. 4·6+5·3= 39; 2·6+7·3= 33. Report 6. 2. 5 2 +7 2 + k = 20 p Þ p = 2, k = 8 2 Þ k =128. Report 128. 3. y = (2 /3)x+ 2Þ 3x - 2y = -6. Report -6. 4. Multiply through by 280 to get 112 < 40k < 245, from which we get k = 3,4,5,6.

Report 18. 5. 2 *(27+ 4)= 2 *31= 4+31= 35. Report 35. 6. c = 2d = (3/4)bÞ b = (8 /3)d Þ b= (8 /3)(3/10)a = (4 /5)a. Report 4/5. 7. 0 = (x - s)(x - s- 2)= x2 - (2s+ 2)x + s2 + 2s = 0. This gives s2 + 2s = 24, from

which s = 4, -6, and k = 2s+ 2 =10, -10. Report 10. 8. (x+3y)2 =10 = x2 +6xy+9y2 =10. This makes 6x2 +36xy+54 y2 = 60. Report

60. 9. The red-yellow-blue triples are 211, 311, 411, 511, 611, 322, 422, 522, 622, 433,

533, 633, 544, 644, 655. There are 15 of these so the probability is 15/216 = 5/72. Report 5/72.

10. 9

x+1 = 32x+2 = 33, so 2x+ 2 = 3 and x = 1/2. Report 1/2. 11. Let the original number be

htu =100h+10t +u =100h+10(u+ 4)+u =100h+11u+ 40. The second number is

100u+10t + h =100u+10(u+ 4)+ h =110u+ h+ 40. Subtract this from the original to get 99h-99u =198, or h = u+ 2. The possible original numbers are thus 351, 462, 573, 684, 795. Report the sum 2865.

12. The side of the square is 6. The rectangle has sides 9 and 6. Report the area 54.

13.

G

R=

R

EÞ R2 =GE =160000 Þ R = 400. Report the total 800+ 200+ 400 =1400.

14. Let f (x) = ax + b. Then 7 = 2a+ b and -5= -4a+ b. Subtract to get 12 = 6a.

Thus a = 2 and b = 3. Then f (-2)= -4+3= -1. Report -1. 15. The given matrix equation gives a = 8, c = -8, k = 4, d =11. Report 32 +88 = 120.

I.C.T.M. 2018 Regional Algebra I – Divisions 1A, 2A

16. Multiply the first equation by 5 and subtract the second to get 13 x = 26, from which x = 4. Then y = 9-6 = 3. Report (4, 9).

17. Cancel the parenthesized terms to get x+1= -1. Report -2. 18. Let x be the original number. Then (6 /5)x - (2 /3)(6 /5)x =120 = (2 /5)x. Report

300. 19. Simplify to get x < 6+ k. Since the largest integer solution is 18, k must be 13.

Report 13. 20. The equation is x3 -5x2 -9x+ 45= 0. Either use trial and error, hoping for

integer solutions, and get 5, 3 and -3. Or solve on your calculator. The possible equations for y, using 1 for a, are (y -5)(y -3) = y2 -8y + 24 = 0,

(y -5)(y+3) = y2 - 2 y -15= 0, and (y -3)(y+3) = y2 +0 y -9 = 0. The probability that b < 0 is thus 2/3. Report 2/3.

I.C.T.M. 2018 Regional Geometry – Divisions 1A, 2A

I.C.T.M. 2018 Regional Geometry – Divisions 1A, 2A 1. (1/ 2)bh = b2 Þ h = 2b. Report 2. 2. ÐFDC = 22ÞÐBFC = 68. Report 68. 3. Let the parallel to BC through D meet AB at E. This creates isosceles triangle

ADE with AD = AE = 9 and EB = 6, making AB =15. Report 15. Alternatively, if the problem is solvable, then the measure of angle B does not matter, so take it to be 45. The rest is easy.

4. Reflect A across the top cushion to get A*. Then AP+ PB = A* P+ PB, which is

minimal when A*, P, B are collinear. Then AB is the hypotenuse of a right triangle with legs 18 and 80, so AB = 82. Report 82.

5. Similar triangles give

x+6

x=

5+3

5, or

1+

6

x=1+

3

5, so x = 10. Report 16.

6. KR = 3MT Þ x2 + 2x -18= 3(x+ 4)Þ x2 - x -30 = 0, so x = 6. Since TS = 3MT ,

TS = 30. Report 30.

7. Let RB = x and ST = y. Then 4x = 3y and y = (4 /3)x. The area ratio

ARE

TSE is

thus Let the area of triangle EST be 16w. Then the area of

triangle ERA is 81w. Let the altitude to ST of triangle EST be z. Then the altitude to AB of triangle ERA is (9 /4)z and the altitude to DT of triangle ADT

is (13/4)z. Since S is the midpoint of DT , the area ratio

ADT

EST is

(1/2)(13z / 4)(2ST )

(1/2)z(ST )=

13

2, so the area of triangle ADT is

13

2(16w) =104w.

Subtract to get the area of quadrilateral AESD to be 88w. The area of triangle RET is 52w-16w= 36w. The required ratio is thus 88w/36w = 22/9. Report 22/9.

8. The arcs are 86, 62 and 212, so angle UAN is 106. Report 106.

9. Let the first polygon have n sides. Then (n- 2)180 = 36

360

n+ 4Þ n- 2 =

72

n+ 4,

leading to n2 + 2n-80 = 0, so n = 8. The number of diagonals in a convex

octagon is 5+5+ 4+3+ 2+1= 20. Report 20.

10. From

10

25=

AD

10 we get AD = 4. Report 4.

I.C.T.M. 2018 Regional Geometry – Divisions 1A, 2A

11. Choose a coordinate system with the leftmost common vertex O being (0, 0) and the rightmost being (4, 0). Then the upper left corner of the hexagon is A(1, 3). Let the top vertex of the square be M (2,2). Let OM and the top edge of the

hexagon meet at E( 3, 3). Then the area of triangle OAE is

(1/2)( 3 -1)( 3)=

3- 3

2. There are four such congruent triangles, whose total

area is 4(3- 3) /2. The area of the hexagon is 6·

22 34

. Divide this by the

previous value to get .244 and report 24.4. 12. Jake went 7 kilometers east and 24 north, ending up 25 kilometers from his

starting point, the last part taking 3/2 of an hour. He thus traveled 56 kilometers in 4 hours for an average speed if 14 kph. Repot 14.

13. Enclose the quadrilateral in a rectangle with A and C on vertical sides and B and

D on horizontal sides. The area of this rectangle is 3·9 = 27. From this subtract the areas of four right triangles outside the required quadrilateral but inside the rectangle. These have areas 2, 3, 3 and 5. What’s left is 14. Report 14.

14. From 2(2x -10)2 = 2(x+ 2)2 we get 3x2 -44x+96 = 0. This factors to give the

roots 12 and 8/3. Only 12 works, so AC = 24 – 10 = 14. Report 14. 15. Complete squares to get (x+3)2 + (y - 4)2 = 36. Report (-3, 4, 6). 16. The other leg is 18, so we get (80 - r)+ (18- r)= 82, so r = 8. Report 8. 17. 3·RK = 9·5Þ RK =15. Report 15. 18. Set up coordinates so that A= (160,0), B = (0,120), C = (0,0). Then M = (80,60)

and an equation of OM is y -60 = (4 /3)(x -80). This meets the x-axis at

K(35, 0), so KM = 452 +602 = 75. Report 75. 19. The areas of ABCD, ABP and BQC are 8, 2 and 2, so the area of BPDQ is 4.

Report 4. 20. Let x be the distance between chords and let y be the distance from the center to

AB. Then r2 = y2 +102, r 2 = ( y+ x)2 +82, r 2 = (y + 2x)2 + 42. Simplify to get

2xy+ x2 = 36, 4xy+ 4x2 = 84. Then x2 = 6 and y =

30

2 6 and

r2 =

225

6+100 =

825

6. Thus

r =

5 22

2. Report 5 + 22 + 2 = 29.

I.C.T.M. 2018 Regional Algebra II – Divisions 1A, 2A

I.C.T.M. 2018 Regional Algebra II – Divisions 1A, 2A 1. 7k +3= 6k Þ k = -3.

2.

k - 4

5+1= -

3

2Þ k = -5

3. 70d = -210Þ d = -3Þ a1 =14. The required sum is

(14-3n) =n=0

776

å 14-3n=0

776

å nn=0

776

å

=14·777 -3

776·777

2= -893550.

4. g = 6·12 = 6 2. a = 9.

g

a=

2 2

3.

5.

6. (2n-1)(2n+1)(2n+3)= 3(2n+1)2 Þ 4n2 + 4n-3= 6n+3, so n = -1 and the

integers are -3, -1, 1. Report -3. 7. Use (0, 1/3) from the first line and use the point-line distance formula to get

d =

2·0+ (-3)(1/3)+5

22 +32=

4 13

13. Report 30.

8. f (4) = k =11.

9. Let x = t. Then y =

20- 4t

5 and

xy =

20t - 4t2

5=-4(t -5/2)2 + 25

5. The maximum

value of xy is thus 5 when x = t = 5/2. Report 5. 10. Square to get x2 - 2x -8 = 0. Then 4 2 0x x- + = so solutions are 4 and 2- .

Report sum of 2. (or use sum of the roots of 2 0ax bx c+ + = is /b a- , so

2 /1 2- - =

11. x =

64zw3

y=

64x2x6

x4 = 64x4 Þ x3 =1

64Þ x =

1

4.

12.

y =

4x-

1x+ 23x2

· x2 (x+ 2)=4x(x + 2)- x2

3(x+ 2)=

3x2 +8x

3x+6. Report 20.

I.C.T.M. 2018 Regional Algebra II – Divisions 1A, 2A

13. The discriminant k2 -4·3·27 = 0, so k 2 = 324 and k = 18.

14. The first base, b, is at least 7. Solve 2b2 +6b+ 4 =1(b-5)3 +3(b-5) to get b = 12.

The base ten numeral is 2·122 +6·12+ 4 = 364. 15. s = 8r and 50 = 8r2 Þ r = 5/2. Thus k =8+ 20+50 = 78. 16. 1620 = 3·540, so six masons could tile 1620 square feet in 12 hours and nine

masons could do that in 8 hours. Report 8. 17. r(r +1)= 306Þ (r +18)(r -17) = 0, so r = 17 and the sum of the roots is 35. 18. log2 (log3(log5 x))= 0Þ log3(log5 x) =1Þ log5 x = 3Þ x =125. log5(log2(log3 y))= 0Þ log2 (log3 y) =1Þ log3 y = 2Þ y = 9. log3(log5(log2 z))= 0Þ log5(log2 z)=1Þ log2 z = 5Þ z = 32. Report 166. 19. a- b+ c = 0, 9a+3b+ c = 0Þ b = -2a, c = -3a. Since c = -9, report (3, -6, -9). 20. The solution region is the triangle with vertices (4, 2), (4, 1) and (10/3, 5/3). The

horizontal altitude is 2/3 and the corresponding vertical base is 1, so the area is 1/3.

I.C.T.M. 2018 Regional Pre-Calculus – Divisions 1A, 2A


1.

4

1- r=10Þ r = 3/5

2. Expand to get

sec2 x -1

sin2 x=

tan2 x

sin2 x=

1

cos2 x= sec2 x, so report sec x.

3. 3x+ 4y = 3 and 2x+3y = 4. Subtract to get x + y = -1 and report -1. 4. Multiply each equation by r to get r 2 = 6r cosq and r 2 = 2r cosq, which convert

to the rectangular equations x2 + y2 = 6x and x

2 + y2 = 2x . These are equations of circles with centers (3, 0) and (1, 0) and radii 3 and 1. The second circle is inside the first so the area between them is 9p -1p = 8p . Report k = 8.

5. d = c- a -b = (36-6-14, 43- 4-9)= (16, 30). 6. Raise each side to the 36-th power to get 26 ·34 = k = 5184.

7. tanq =

m2 -m1

1+m2m1

=5-1/4

1+5/4=

19

9Þq = 64.7 or 115.3. Report 115.3.

8. 11-60i = 121+3600 = 61. 9. b = 4, so [k, w]= [3- 4, 3+ 4]= -1, 7[ ]. Report 6. 10. The dot product is -12+3k, which is 0 when k = 4. 11. When A,B are chosen, the probability of the same color is

C(3,1)C(2,1)+C(6,1)C(1,1)

C(9,1)C(3,1)=

6+6

27=

12

27. When A,C are chosen, the probability

of the same color is

C(3,1)C(2,1)+C(6,1)C(4,1)

C(9,1)C(6,1)=

6+ 24

54=

15

27. When B,C are

chosen, the probability of the same color is

C(2,1)C(2,1)+C(1,1)C(4,1)

C(3,1)C(6,1)=

4+ 4

18=

12

27. Each of these must be multiplied by

1/3, the probability of the box choice, so the result is 13/27. 12. The amplitude is 3. 13. 25/ d = sin52Þ d = 31.7.


14. Solve (4sin x+1)(3sin x -1) = 0 to get sin x = -1/4 or 1/3. Graph y1(x) = sin x,

y2(x)= -1/4 and y3(x) =1/3 to see intersections at sin-1(1/3), p - sin-1(1/3),

2p + sin-1(-1/4), p - sin-1(-1/4). The sum of these is 4p , so report 4.

15.

204

5p = 40p +

4

5p , so k = 4/5.

16. The period of f is

2pk

and the period of g is

pw

. These are equal when k =±2w.

Since k +w= 6, the (k, w) possibilities are (4, 2) and (12, -6). Report 16. 17. The Law of Cosines gives 100 = 288- 288cosq, so cosq =188 /288 and

q = .859551radians. The arc length is s = rq = (12)(.859551) =10.3. 18. a = (k + w) /2 and g = kw. Then (k +w) /2+ kw =18Þ k + 2 kw +w= 36, but

k + 2 kw +w = k + w( )2

, so k + w = 6.

19. as , so the limit is 4/5. 20. Let p = (-420sin40, 420cos40) be the vector corresponding to the plane with no

wind. Let w= (-20sin20, - 20cos20) be the wind vector. The resultant vector representing the plane’s actual path is

r = (-420sin40- 20sin20, 420cos40- 20cos20) = r = (-276.81, 302.94). The

angle q for this vector is Report 42.42.

I.C.T.M. 2018 Regional Freshman-Sophomore 8-Person – Divisions 1A, 2A

I.C.T.M. 2018 Freshman-Sophomore 8-Person – Divisions 1A, 2A 1. Multiply the second equation by 8 and subtract to get 19x = 76. Report (4, 2).

2. 71 and 83 are primes, so the probability is 2/5. 3.

The distance is 61. 4. 15¸5·3= 9 and 9-3/2 =1/27. 5. (13+ k ) /4 = 8Þ k =19. The range is 19 – 2 – 17.

6. Simplify to

x + 3x + 9x + 27x = 4+ 4 3( ) x . Report 4 + 4 + 3 = 11.

7. Let AE = 1 and EB = x. Then

1+ x

1=

1

xÞ x2 + x -1= 0Þ x =

-1+ 5

2Þ1+ x =

1+1 5

2. Report 1+1+5+ 2 = 9.

8. 2A=

kB2

4CÞ 4A C = kB2 , so

2 A=

kB2

4CÞ 4·2 9 = 36k Þ k =

2

3. Then

C =1/6 and C =1/36. 9. The sequence of remainders is 3, 4, 3, 4, ..., so report 4. 10. 27x3 + 27x2 +9x+1-9x2 -6x -1= 27x3 +18x2 +3x+0. Report 48.

11. (x+7)2 + ( y -3)2 = (x+1)2 + (y -3)2 + 4. Square and collect terms to get

12x+32 = 8 (x+1)2 + (y -3)2 . Repeat to get 5x2 - 4 y2 + 40x+ 24 y + 24 = 0. Report 89.

12. Draw the parallel to AB through F and get ÐFED = 90. 13. 5-1 + 2-2 =1/5+7 /4 = 9 /20. 14. Subtract the third from the second to get 5y - k - 4. Subtract 3 times the third

from 2 times the first to get (2k -3)x+6y = 0. Subtract the first from k times the

second to get (k +3)y = k 2 -6. Then . This factors into

(4k +9)(k - 2) = 0, so k = 2 or -9/4. Report k = -1/4.


15.

nn=1

10

å = 55Þ 55- 2x = 41Þ x = 7

16. f (4 /3) = 6(4 /3)2 -5(4 /3)+ k = -10Þ k = -14. 17. There are 27: 6,8,10,12,14,15,18,20,21,22,24,26,27,28,30,32,33,34,

35,38,39,40,42,44,45,46,48. Report 27. 18. 3·32x = 381 Þ 2x+1= 81Þ x = 40 19. P(no 6)= 25/36. P(sum = 6)= P(15,51,24,42,33) = 5/36. Report 5/36 + 11/36 = 4/9. 20. Start with meal A: 1 day gives 1, 2 days give AA or AB; 3 days give AAB or

ABA or ABB; 4 days give AABA or AABB or ABAA or ABAB or ABBA; 5 days give AABAA or AABAB or AABBA or ABAAB or ABABA or ABABB or ABBAA or ABBAB, and so on. The numbers here form the sequence 1, 2, 3, 5, 8, which suggest the 10-day totals starting with A: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89. The same sequence applies if we start with B, so the total is 2·89 =178. Report 178.

I.C.T.M. 2018 Regional Junior-Senior 8-Person – Divisions 1A, 2A

I.C.T.M. 2018 Regional Junior-Senior 8-Person – Divisions 1A, 2A 1. 243000 = 92.5 ·53 ·23 Þ log2 23000 = 2.5log2 9+3log2 5+3log2 2 = 3x + 2.5y +3.

Report 8.5. 2. x

2 - x = 252. Guess 16 and see that it’s right. Report 16.

3. Let A be the denominator. Then A=1+

7

AÞ A2 - A-7 = 0, so

A=

1+1 29

2.

Report 33. 4. Reduce to 15x2 -30x+10x2 -40 = 6x2 +6x -36, or

19x2 -36x - 4 = 0 = (x - 2)(19x+ 2). Report the sum 2- 2 /19 = 36 /19. 5. The first equation gives xy = 0, so 4xy = 40 =1. 6. Let the roots be r - d, r, r + d. Then 3r = 15 and r = 5, and

r 3 - rd 2 = 80Þ125-5d 2 = 80Þ d = 3. The roots are 2, 5, 8; use 2 to get k = 66. 7. tan 1 is a bit less than 3. The others are less than 1.1. Report tan 1. 8. The only solution is m = 4, p = 2. Report 256.

9. . . Subtract

the second equation from the first to get .

Multiply by 2 to get

10. tan x+

1

tan x= 4Þ

sec2 x

tan x= 4Þ

1

sin x cos x= 4Þ

1

2sin x cos x= 2

Þ sin2x =

1

2.

11. Let p be the radius of circle P and let ÐPQR =q. Apply the Law of Cosines to

triangle PQR to get (1+ p)2 = 32 + (2+ p)2 - 2·3· (2+ p)cosq. Apply to triangle PQO to get (3- p)2 =12 + (2+ p)2 - 2·1·(2+ p)cosq . Multiply the second equation by 3 and subtract from the first to get

(1+ p)2 -3(3- p)2 = 9-3- 2(2+ p)2. Reduce to -26- 20 p = -2-8p, so p = 6/7.

12.

1

1+1/ x= 2Þ1+1/ x =1/2Þ x = -2.

13. Multiply numerator and denominator by 2 - 3 to get 6 - 2.


14. There are 5 1-digit odd numbers. For 2-digit odd numbers there are 5 ways to fill the units place and 8 ways for the tens place, making 40. For 3-digit numbers there are two cases: with 0 in the tens place there are 8·1·5= 40 numbers; with non-zero tens digit there are 7·8·5= 280 numbers. For 4-digit numbers there are three cases, depending on where/whether 0 appears: 7·8·1·5= 280, 7·1·8·5= 280, 6·7·8·5=1680, for a total of 2240 4-digit numbers. The grand total is 2605.

15. Cancel to reduce to 99x = 450- x , so x = 9/2.

16. since the process leaves all of these

integers remaining just one time. To analyze SS(B) look at a smaller case with n = 10 and the numbers being 1 through 100. With the selections being random, it can’t matter if we use the number on the main diagonal: 1, 12, 23, 34, 45, 56, 67,

78, 89, 100. The sum of these can be written as

(1+11k) =10+11·9·10

2=

k=0

9

å 505.

With n = 100 we get, again using the main diagonal,

(1+101k) =100+101·

99·100

2=

k=0

99

å 500050. The difference is 495000.

17. Factor and reduce to get x

3 + x2 + x +1, which approaches 4 as x approaches 1.

18. The expression factors to

x2 -9( ) x2 +9( )(x +6)2 , whose sign depends on the sign of

x2 -9, which is negative at -2, -1, 0, 1, 2. The sum of these is 0. 19. Tan-1(-1) = -p / 4, so report -1/4.

20. Use partial fractions:

3

n2 - n- 2=

A

n- 2+

B

n+1. Clear fractions to get

3= A(n+1)+ B(n- 2). Unlike the given rational expression, this equation is valid when n = 2 or 1. With n = 2 we get A = 1 and with n = -1 we get B = -1. The sum

is then . The only fractions that

don’t get canceled are

1

1,

1

2 and

1

3. Report the sum

11

6.

I.C.T.M. 2018 Regional Calculating – Divisions 1A, 2A

I.C.T.M. 2018 Regional Calculating – Divisions 1A, 2A 1. Solve with “and” between equations to get (46.51, -7.735). 2. Use a table, with y1(x)= (ln x) /(ln17), starting with x = 1000 and steps of 1000 to

narrow x between 4000 and 5000. Next start with 4000 and steps of 100 to get x between 4900 and 5000. Finally, start at 4900 with steps of 1 to get 4913 that produces 3.0000.

3. With n = 20 or 40 we get the mean and median to be 13.45 and 13. Report .45.

4.

500(12.87-12.75)

12.87= 4.66, the exchange fee in USD.

5. F(21)=10945.999981729 and F21 =10946, so report 1.827´10-5. 6. Solve 5000(1+ (r /100)/12)12t = 5230 for t to get 4.506. 7. The triangle area formula involves sinq , which equals sin(180 -q). These are

greatest, 1, when q = 90. The sum of the areas of the four triangles is then

(1/2)(10·17+10·37+ 27·17+10·37) = 999. 8. Define f (x)= x2 +0.2+0.1i. Evaluate f (0.1+0.3i) and then do f ( Ans) five

times to end up with .22 + .18i.

9. Solve (x2 + x+ 2)/ x = y for x to get two answers involving y2 - 2y -7 . This

must be real in order to get the two answers in the range. Solving y2 - 2y -7 = 0

gives 3.828427 and -1.828427. Numbers between those two do not appear in the range, so report (-1.828, 3.828).

10. P(3M ,0W ) =

C(5,3)

C(11,3)=

10

165 and

P(0M ,3W ) =

C(6,3)

C(11,3)=

20

165. The sum of these,

30

165, is the probability of a single-gender committee. Subtract from 1 to get 9/11

for the probability of at least one of each gender. Report 9/11. 11. The radius of the outer circle is p(9 /2)2 . Each of the four quarter-circles has

radius 3.5- 2 /2, so the sidewalk area is p (9 /2)2 - 3.5- 2 /2( )2[ ] =12.45p.

Report 12.45. 12. The fourteenth term is 1.934563, as are all subsequent terms on the calculator, so

report 1.935 for the 2018-th term.

I.C.T.M. 2018 Regional Calculating – Divisions 1A, 2A

13. Solve or note that 27x = -57 /2Þ x = -19 /18. 14. The inside sum is 1+ 4+9 =14, so the outside sum is 14 times (1 + 2 + 3 + 4) =

140. 15. The sum is -e /3= -.9061.

16. The amplitude of y = Asinq + Bcosq is A2 + B2 = 8.715 here. 17. Only powers of 3 and 7 can end in 7. Trial gives 57 since 575 = 601692057,

fitting the requirement. No other integer fits. Report 57.

18. Take ln of both sides to get xp xp...

= lnp . Take ln of both sides to get

ln xp( ) ln p[ ] = ln(lnp )Þ (lnp )ln xp( ) = ln(lnp )Þ ln xp( ) = ln(lnp )

lnp= .118079.

Then xp = e.118079 =1.125333, from which

p ln x = ln(1.125333)Þ ln x =

ln(1.125333)

p= .037586, so x = e.037586 =1.038.

19. , which is the gross value to the person receiving the rebate.

The net value to the community is thus 2500- 500 = 2000. 20. Let AB = x. The Law of Cosines gives 62 = 82 + x2 -16x cos48. Solve to get x =

6.1624 or 4.5437. Use Area = (1/ 2)(8x)sin48 to get the two values 13.506 and 18.318. Report the sum of these, 31.82.


I.C.T.M. 2018 Regional Freshman-Sophomore 2-Person – Divisions 1A, 2A 1. 67 – 27 = 40 2. Expand to get 3x2 +8x +7 and report (3, 8, 7). 3. The first equation gives x = 15. The second gives y = 20. The hypotenuse is 25. 4. 5(x -6)= 3x -12Þ x = 9. 7(y - 2)= 4(y+1)Þ y = 6. The side are 15, 15, 28 and

28 and the perimeter is 86. 5. A= (4, 2) and B = (1, 6). Then AB = 9+16 = 5.

6. r =

20

pÞ K = p ·

80

p= 80 and

r =

20

pÞW = p ·

5

p= 5. Report 75.

7. K =12, W = 6. The arithmetic mean of these is 9. 8. Look for a pattern. When n = 1, the number S of squares is 1. When n = 2, the

number S of squares is 1 + 4 = 5. When n = 3, the number S of squares is 1 + 4 + 9 = 14. When n = 4, the number S of squares is 1 + 4 + 9 + 16 = 26. Keep adding squares until the 14-th gives a total of 1015. Report 14.

9. 2x+ 2x+3x+3x+5x = 360Þ x = 24. The largest exterior angle is 120, so K = 60. 2y+3y +7 y =180Þ y =15. The smallest interior angle is 30. The third angle of

the final triangle is thus 90. 10. The possible third sides are 9, 10, 11, 14, 15, 16, 17. Thus N = 7.

2x -3 < 9Þ-9 < 2x -3< 9Þ-3< x < 6. Then x = -2, -1, 0, 1, 2, 3, 4, 5 and W = 8. Report 15.


I.C.T.M. 2018 Regional Junior-Senior 2-Person – Divisions 1A, 2A 1. (a,b,c,d) = (1,-3,3,23). 43/2 = x =8. Report 32. 2. 161.2( ) 16-.7( ) = 16.5( ) = 4

3. f (6)+ g(-3) = 21- 4 =17 4. BC =18, so cos A= 24 /30 and cos B =18/30 cos B : cos A=18 :24 = 3:4. 5. The denominator factors as (3x+ 4)(x -1), so a =1, b= -4 /3. Divide all terms by

x2 to get

2-7 / x -15/ x2

3+1/ x - 4 / x2 to see that c = 2/3. Report -1.

6. The sums are

7+9+11+13+15+17+19

1-3/4= 4·3·26. The average is 52.

7. The axis of symmetry of the parabola is x = 4, so P = (4,6). When x = 2 on the

circle, y = -2± 4. Then PQ = 4+144 = 2 37, 4+16 = 2 5. Report 2 37 . 8. a20 = 4·192 +58 =1502 and a18 = 4·172 +52 =1208. Report 294. 9. The roots are 0, -3, -2, 2, 3. Only -3 fails the inequality, so report 4/5. 10. The term referred to is C(5,3)(2x)3 y2 = 80x3 y2. An equation of the circle is

(x - 2)2 + (y+5)2 = 7+ 4+ 25= 36, so d = 12. Report 82 + 12 = 94.

27°

16C B

A

ORAL COMPETITION PAGE 1 OF 2 ICTM REGIONAL 2018 DIVISION A JUDGES’ SOLUTIONS 1. Given ABC with C 90 , B 27 and BC 16 . Find the length of the hypotenuse of

this triangle. Give your answer rounded to the nearest hundredth.

SOLUTION: 16 16

cos 27 ABcos27 16 AB 17.96AB cos 27

or students may find the measure of A 63 , and use Law of Sines

2. A 15-foot long ladder is leaning against a building. The top of the ladder is 8 feet off the ground.

If the top of the ladder slides 3 feet down the building, what is number of degrees in the change in the angle of elevation at the base of the ladder? Give your answer in degrees/minutes/seconds form, rounded to the nearest second.

SOLUTION: Let represent the original angle of elevation, and let represent the angle of

elevation after the ladder slid down the wall.

1

1

8 8sin sin

15 15

8 3 5sin sin

15 15

so the change in

the angle of elevation is 1 18 5sin sin 12 45'35"

15 15

3. The Bermuda Triangle has sides whose lengths are approximately 850 miles, 925 miles, and

1300 miles. Find the degree measure of the largest angle of the Bermuda Triangle. Round your answer to the nearest tenth of a degree.

SOLUTION: The largest angle is opposite the longest side. Using the Law of Cosines,

2 2 2

2 2 2 1300 850 9251300 850 925 2 850 925 cos cos 0.0711 94.1

2 850 925

ORAL COMPETITION PAGE 2 OF 2 ICTM REGIONAL 2018 DIVISION A JUDGES’ SOLUTIONS 4. Harry is playing the 17th hole on a golf course. The hole’s length from tee (T) to the pin (P) is

182 yards, and includes hitting over a large lake. Harry can’t hit the ball that far, and decides to play around the lake. On his first shot, he hit the ball 170 yards on a bearing of N 60 E to point B. On his second shot, he hit the ball on a bearing of N 157 E and the ball went in the hole at the pin (P). Find the measure of BPT . Round your answer to the nearest degree.

SOLUTION:

METHOD 1: Since all north-south lines are parallel, and consecutive interior angles of parallel lines are supplementary, a 120 angle exists as shown:

This makes PBT 360 157 120 83 METHOD 2: Again, since all north-south lines are parallel, and alternate interior angles of

parallel lines are equal, a 60 angle exists as shown. Using the north-south line at point B, if the exterior angle to the triangle is 157 , the interior angle is 180 157 23 . This makes

PBT 60 23 83

From either method of finding the angle at B, using the law of sines, sin83 sin P

P 68182 170

170

182

B

T P

170

182

B

T P

24

40°C B

A

θ10

6

ORAL COMPETITION PAGE 1 OF 2 ICTM REGIONAL 2018 DIVISION A JUDGES’ SOLUTIONS EXTEMPORANEOUS QUESTIONS Give this sheet to the students at the beginning of the extemporaneous question period. STUDENTS: You will have a maximum of 3 minutes TOTAL to solve and present your solution to these problems. Either or both the presenter and the oral assistant may present the solutions. 1. Using the given diagram,

Find the value of sin , cos and tan SOLUTION: The sides of this triangle are multiples of a 3-4-5 triangle, so the missing side length

is 8 (students may also use Pythagorean Theorem to find this side).

8 4 6 3 8 4sin ; cos ; tan

10 5 10 5 6 3

2. Given triangle ABC as shown, set up equations for two

different ways you could find the length of BC . You do NOT need to do the calculations to find the length.

SOLUTION: Most students will probably use BC

cos 4024

and BC

sin 5024

. These two

equations are sufficient, although students may rewrite them to solve for BC. Any other valid equation should also be accepted; the most likely would be using the law of

sines: sin 90 sin50

24 BC

, although other methods are possible such as solving for AC and

using the Pythagorean Theorem.

ORAL COMPETITION PAGE 2 OF 2 ICTM REGIONAL 2018 DIVISION A JUDGES’ SOLUTIONS EXTEMPORANEOUS QUESTIONS 3. In each scenario for ABC listed below, comment on whether it is possible to draw two different

triangles with the given information. a. BC 7.0, A 110 , C 33 b. BC 23, AB 11, A 122 c. BC 8.1, AC 8.3, A 42 SOLUTION: Two solutions may exist when the given information is SSA.

a. In this triangle, two angles are given. There cannot be two triangles as the third angle must be 180 110 33 37 and one side is given. b. While the given information is SSA, the angle given is obtuse. There cannot be two different triangles, since the other angles must be acute. c. There could be two triangles with this given information.

sin 42 sin B

B 43 or 1378.1 8.3

so there are two triangles with this given

information. (Note that students do not necessarily need to find the possible angles, if they explain that it would be possible since the only requirement is that angle B is larger than 42 and so may be either acute or obtuse)

WRITTEN AREA COMPETITION ALGEBRA IICTM REGIONAL 2018 DIVISION AA PAGE 1 OF 3

1. A number is tripled and the result is decreased by 8. The result is a value that is 2 more thantwice the original number. Determine the original number.

2. Determine the value of k for which ( )( )50 70k = .

3. The slope of a line is2

3and the intercepty − of the line is 2. An equation for this line can

be written in the form Ax By C+ = where A , B , and C are relatively prime integers with

0A > . Determine the value of C .

4. The larger of two positive integers less than 1000 is 3 more than the smaller. The largerinteger is a multiple of 47, and the smaller is a multiple of 23. Determine the sum of the twointegers.

5. The price of a certain stock increased by 60% at the end of the first year of ownership. At

the end of the second year, its value was5

6of what it was at the beginning of that year.

During the third year, it lost 15% of its value. At the end of the third year, the stock pricewas $272. Determine the initial value of the stock at the beginning of the first year. Expressyour answer rounded to the nearest whole dollar.

6. Determine the value(s) of x such that 16 2 3x x+ =

7. One solution for x is 2 more than the other solution for x in 2 24 0x kx+ + = . Determine thelargest possible value of k .


8. Let ( ) 2 3f x ax kx= + − with ( )4 25f = and ( )7 88f = . Determine the ordered pair ( ),a k .

Express your answer as that ordered pair.

9. A red, yellow and blue die are tossed and the number facing up on each die is recorded.Determine the probability that the yellow and blue die show the same number and the sum ofthe numbers on all three dice is less than 9. Express your answer as an integer or as acommon fraction.

10. Determine the value(s) of x for which

23 5 2.5251

5

x x+ +

= . Express your answer(s) as an integer

or as a common or improper fraction.

11. A positive three-digit number has the property that, when the digits are reversed, it forms asecond three-digit number. When the second number is subtracted from the original number,the difference is 198. In the original number, the tens digit is 4 more than the units digit.Determine the sum of all such possible original numbers. (Note: 012 is not considered athree-digit number.)

12. An equilateral triangle has a side of length 8. A square has the same perimeter as theequilateral triangle. The length of one pair of opposite sides of the square is increased by 3and the length of the other pair of sides is left as is. Determine the numeric area of therectangle formed.

13. In a recent election for mayor, there were 3 candidates, Riemann, Euler and Gauss. The ratioof Gauss voters to Riemann voters is the same as the ratio of Riemann voters to Euler voters.There are 4 times as many votes for Gauss as there were for Euler and there are k times asmany votes for Riemann as there are votes for Euler. Each voter voted for one and only oneof the three candidates. Determine the value of k .

14. There are 2018 balls in an urn, all of which are either red, white or blue. The ratio of redballs to white balls is 5 to 3 and there are 182 more white balls than blue balls. Determinethe number of balls in the urn that are red.


15. In a right triangle, the length of the hypotenuse is 3 less than twice the length of the shorterleg, and the length of the longer leg is 8 less than the length of the hypotenuse. Determinethe number of square units in the area of this right triangle.

16. Determine the value of ( )y x− when 1 2 1x y+ = − − and 2 2 1 1y x− = + − .

17. The points ( )5,40 , ( )3,32− , W and P lie on the graph of 2y ax bx c= + + . Point W has

coordinates ( )3,k and point P has an coordinatex − of 2 and a coordinatey − that is 1 more

than 3 times the value of a . Determine the value of k .

18. Twenty-four students are divided into 3 groups. There are 4 more students in group II thangroup I and twice as many students in group I than group III. Determine the number ofstudents in group II.

19. The largest integral solution for x in ( )3 2 2 4x x x k− + < + is 18. Determine the smallest

possible integral value of k .

20. Two of the solutions for x in the equation 3 29 5 45x x x− = − are selected at random. These

two values are the solutions for y in the equation 2 0ay by c+ + = where a , b , and c are

integers with no common factors and 0a > . Determine the probability that 0b < . Expressyour answer as a common fraction.

2018 RAA Name ANSWERS

Algebra I School(Use full school name – no abbreviations)



1. 11.

2. 12.

3. 13.

(“red balls” or“balls” optional.)

4. 14.

(“dollars” or “$” (“square units” oroptional. Accept 240.00.) “sq. un.” optional.)

5. 15.

(Must be thisvalue only.)

6. 16.

7. 17.

(Must be thisordered pair.) (“students” optional.)

8. 18.


9. 19.

(Must have both answers, (Must be this reducedIn either order.) common fraction.)

10. 20.

279

2

3

13

12

14

8

630

1000

2

54

2865

1− , 2

3− OR

2

3

−

1

18

( )2, 1−

10

4

240

98

6−

10

WRITTEN AREA COMPETITION GEOMETRY ICTM REGIONAL 2018 DIVISION AA PAGE 1 OF 3

1. Five points are arranged so that MT TC MC , T and C trisect HM , A bisects TM , 20CM , 2AT y , and 3 1HC x . Determine the ordered pair ,x y . Express your

answer as that ordered pair ,x y

2. The altitude to the hypotenuse of a right triangle divides the hypotenuse into segments of

length 3.2 and 14.45 . Determine the area of this right triangle. Express your answer as an exact decimal.

3. ABCD is a trapezoid. Leg 9AD , base 6CD and 2m D m B .

Determine the length of base AB . 4. The line 2 3x y C is tangent to the circle 2 22 4 95x x y y . There are two values

possible for C . In reduced form, the positive value of C may be written as k w p

Cq

.

Determine the sum k w p q .

5. An equilateral triangle is constructed exterior to a square with one side of

the square as base of the equilateral triangle as shown in the diagram. Determine the degree measure of BCE .

6. Square CDEF is inscribed in right ABC with right angle at C . That is, the square and the

triangle share right angle C , E lies on AB , D lies on BC , and F lies on AC . : 5 : 2AC ED and : : 2AC BC k . Determine the value of k .

7. Quadrilateral ABCD is a parallelogram. Q is the midpoint

of AB , R is the midpoint of QB , and S and T trisect

DC . Determine the ratio of the area of quadrilateral AESD to the area of RET . Express your answer as an integer or a common or improper fraction.

B

C

D

A

E

B

C

A

D

E

RQ B

CD

A

S T

WRITTEN AREA COMPETITION GEOMETRY ICTM REGIONAL 2018 DIVISION AA PAGE 2 OF 3 8. A triangle has an inscribed circle of radius 4. Determine the ratio of the perimeter of the

triangle to the numeric area of the triangle. Express your answer as the reduced ratio in the form :k w .

9. In the diagram, C and E trisect BD , F bisects BC , AC AD , 3BF ,

and 4AE . Determine the length of AD .

10. In the circle, chords AB and CD are perpendicular and intersect at point E . 2CE , 6DE , and 3EB . The area of this circle is k . Determine the value of k . Express your answer as an integer or as a common or improper fraction.

11. A square tile sits on top of a regular hexagonal tile so that the diagonal of

the square exactly overlaps one of the longest diagonals of the hexagon. The area of the hexagon that shows from under the square tile is %k . Determine the value of k . Express your answer as a decimal rounded to the nearest tenth and without using the % symbol.

12. Starting at noon, Jake rode his bike due east averaging 14 kilometers per hour. A half hour

later at 12:30, he turned due north and rode at 12 kph until 2:30 pm. At that point, he was able to turn on a straight road that took him directly back to the original starting point, arriving there at 4:00 pm. Determine the average speed in kph for his entire ride.

13. LOG is a right triangle with right O . MA LG , HT LG , AL AM , and HT TG . 4LM , 2OM , 2GH , and

4OH . Determine the numeric area of quadrilateralMATH . 14. The points 0,6 , 3,7 , and 7,5 lie on a particular circle. The area of this circle is k .


F E DC

A

B

C D

A

B

E

A TL G

O

M

H

WRITTEN AREA COMPETITION GEOMETRY ICTM REGIONAL 2018 DIVISION AA PAGE 3 OF 3 15. An equilateral triangle is inscribed in a square symmetric to a diagonal of the

square. Let s denote the length of a side of the square. Then the area of the triangle is 2ks . Determine the exact value of k .

16. Evie and Xavier have agreed to meet at the mall between 1:30 pm and 2:45 pm tomorrow.

Evie will wait at most 15 minutes for Xavier to show up, and Xavier will wait at most 7 minutes for Evie to show up. Determine the probability that they will actually meet at the mall tomorrow. Express your answer as an integer or as a common fraction.

17. Three sides of quadrilateral ABCD are tangent to a circle at points X , Y ,

and Z . 6AB , 7BC , 8CD , and 9AD . Determine the numeric value of the sum of the lengths CY DX .

18. Quadrilateral ABCD is an isosceles trapezoid with EF perpendicular to the bases. 8AB , 34AC , and 30EF . Determine the perimeter of trapezoid ABCD .

19. Quadrilateral ABCD is a rectangle with 4AB and 2BC .

Segments from B to the midpoints P and Q of the two other sides are drawn to form quadrilateral BPDQ . Determine the area of quadrilateral BPDQ

20. AB , CD , and EF are parallel chords on the same side of the center of a

circle. The distance between AB and CD is equal to the distance

between CD and EF . 20AB , 16CD , and 8EF . The radius of

this circle is then expressed as k w

rp

in reduced radical form.

Determine the sum k w p .

Q

P

D

B

C

A

DF

A B

CE

B

A

C

D

Y

X

Z

FC

BA

D

E


Geometry School(Use full school name – no abbreviations)



(Must be this (Must beordered pair.) this decimal.)

1. 11.

(Must be this (“kilometers per hour”exact decimal.) or “kph” optional.)

2. 12.

3. 13.


("degrees" or answer.)

" "° optional.)

5. 15.


6. 16.



(Must be this reduced answer or simplifiedration in this form.) equivalent.)

8. 18.

(Must be thisexact answer.)

9. 19.


10. 20.

4 241 32+ OR ( )4 241 8+

OR 32 4 241+ OR ( )4 8 241+

29

4

( )3,7

10

1513

5625

2 3 3− OR

3 2 3− + OR ( )3 2 3−

25

9

14

24.4

65

4

2 13

1: 2

22

9

3

75

20

15

60.01

WRITTEN AREA COMPETITION ALGEBRA IIICTM REGIONAL 2018 DIVISION AA PAGE 1 of 3

1. Determine the value of 1x− when1

33 1x = . Express your answer as an integer or as a


2. Determine the sum of all possible solution(s) when ( )2log 14 5 2x x− + = .

3. The 7th term of an arithmetic sequence is 4− . The 77th term is 214− . Determine the sum ofthe first 777 terms.

4. Let 6x y+ = and 2 2 10x y+ = . Determine the value of ( )2

x y− .

5. The first term of an infinite geometric series is 12 and the sum of the series is 20. Determinethe third term of the series. Express your answer as an integer or as a common or improperfraction.

6. Consider the compound inequality 2 23 9x y x+ ≤ ≤ − + . The domain of the solutions to this

system is the interval [ ],a b and the range of the solutions to this system is the interval [ ],c d .

Determine the sum ( )a b c d+ + + .

7. The distance between the two lines 2 3 1 0x y− + = and 3 2 5y x− = , when in simplest radical

form, is written ask w

p. Determine the sum ( )k w p+ + .

8. The first three terms of an arithmetic sequence are log 5 , log k , and log 45 in that order.



9. Consider the points ( )6, 2P − and ( )2,10Q . The perpendicular bisector of PQ is a line that

can be written as 0Ax By C+ + = where A , B , and C are relatively prime integers with

0A > . Determine the sum ( )A B C+ + .

10. Let ( )2 1

3

xf x

+= . Then ( )1 kx w

f xp

− += where k , w ,and p are relatively prime integers

with 0p > . Determine the sum ( )k w p+ + .

11. x varies inversely as y and jointly as z and the cube of w . The constant of variation is 64.

Determine the value of x when z y= , 2w x= , and 4y x= . Express your answer as an

integer or a common or improper fraction.

12. When

2

4 1

23

x xy

x

−+= is written as a single simplified rational expression, it is in the form

2ax bx cy

dx e

+ +=

+. Determine the sum ( )a b c d e+ + + + .

13. ( ) 3 2f x x ax bx c= + + + . The sum of the zeros of ( )f x is 5, the product of the zeros is 6,

and ( )2 0f = . Determine the value of b .

14. Determine the sum of all possible value(s) of x such that ( )22log 1 3x x+ + = . Express


15. The graph of the function ( )f x x= is translated 2 units to the right, then reflected over the

x-axis, and finally translated 3 units up. The resulting function is ( )g x k x w p= − + .

Determine the sum ( )k w p+ + .


16. Everett drove from his farm to town at 45 miles per hour on the back road. Going home, hetakes the highway which makes the trip twice as long, but he can increase his speed by 80%.Determine his average speed for the whole trip. Express your answer as an integer or as acommon or improper fraction.

17. ( ),m n with m n> is a solution to the system

3 3

1 14

1 1316

m n

m n

+ = −

+ = −

. Determine the sum

( )m n+ . Express your answer as an integer or as a common or improper fraction.

18. An ellipse has foci at ( )5,0 and ( )5,0− . The major axis is 18 units long. Determine the

length of the minor axis.

19. When graphed, the parabolic equation 2y ax bx c= + + has zeros 1− and 3 and passes

through the point ( )0, 9− . Determine the ordered triple ( ), ,a b c . Express your answer as

this ordered triple.

20. The solutions to the system

5

2

4

0

x y

x y

x

y

+ ≥ ≥

≤ ≥

are graphed and form a polygonal region in the

coordinate plane. Determine the area of this region containing the solutions to the system.Express your answer as an integer or a common or improper fraction.


Algebra II School(Use full school name – no abbreviations)




1. 11.

2. 12.

(Comma usageoptional.)

3. 13.


4. 14.


5. 15.


6. 16.


7. 17.

(Must be thisexact answer.)

8. 18.

(Must be thisordered pair.)

9. 19.


10. 20.

27

1

3

( )3, 6, 9− −

4 14

4

21

1215

19

4

63

16

9

20

1

4

4

6

15

30

12

48

25

16−

893550−

14

WRITTEN AREA COMPETITION PRECALCULUS ICTM REGIONAL 2018 DIVISION AA PAGE 1 OF 3 1. A boy travels along a path given by 5sinr where r is measured in feet. Then the

distance in feet this boy traveled for 0 2 is k . Determine the value of k . 2. An owl was perched on a vertical pine tree with his eyes 24 feet above ground. This owl

spots a mouse on the level ground at an angle of depression of 50 . Determine the straight-line distance from the owl to the mouse. Express your answer rounded to the nearest tenth of a foot.

3. ,k w is the solution to the matrix equation 3 4 3

2 3 4

x

y

. Determine the value of the

sum k w .

4. Let 1

18

3

n

n

k

. Determine the value of k .

5. Let cos ,sinW be the wrapping function around the unit circle. Determine the

straight-line distance between W and 4

W

. Express your answer as a decimal

rounded to the nearest thousandth. 6. Two cruise ships leave the same port at the same time so that the angle between their paths is

60 . The first ship travels at 16 mph and the second ship travels at 14 mph. Determine how long they had been sailing at the time they are 26.424 miles apart. Express your answer as a decimal rounded to the nearest tenth of an hour.

7. Two lines 1

13

4y x and 2

15

2y x intersect. Determine the degree measure of the

obtuse angle formed by this intersection. Express your answer as a decimal rounded to the nearest tenth of a degree.

WRITTEN AREA COMPETITION PRECALCULUS ICTM REGIONAL 2018 DIVISION AA PAGE 2 OF 3

8. Let vectors 6 78u i j

and 117v i k j

such that u v

. Determine the value of k . Express your answer as an integer or as a common or improper fraction.

9. (Always, Sometimes, or Never) For your answer, write the whole word Always,

Sometimes, or Never—whichever is correct. The graph of a certain linear function has a negative slope. Consequently, if its x-

intercept is a negative number, then its y-intercept is also a negative number.

10. Let 1

sin cos3

x x . Determine the value of sin 2x . Express your answer as an integer or

as a common or improper fraction. 11. Box A contains 3 red marbles and 6 blue marbles. Box B contains 2 red marbles and 1 blue

marble. Box C contains 2 red marbles and 4 blue marbles. Two of the boxes are randomly chosen. Then one marble is chosen from each of those two boxes. Determine the probability that these two marbles drawn are the same color. Express your answer as a common fraction.

12. Consider the graph of the function 3sin 2 5f x x . Determine the amplitude of

f x .

13. Let x represent the degree measure of an angle such that tan( ) 3x . If 0 720x , determine the sum of all possible distinct values of x .

14. a , b , and c are positive integers with b c such that 3 3

3

1

2 1a b c

. Determine the

ordered triple , ,a b c . Express your answer as that ordered triple.

WRITTEN AREA COMPETITION PRECALCULUS ICTM REGIONAL 2018 DIVISION AA PAGE 3 OF 3

15. Line a has slope 3

2 and line b has slope 4. Line bisects an angle formed by a and b and

has positive slope. That slope can be written as k w p

q

where k , w , and q are relatively

prime integers and 0q . Determine the sum k w p q .

16. Let T be the tetrahedron bounded by the points 0,0,0 , 0,0,8 , 0,6,0 , and 10,0,0 .

The unit vector V

normal to the face of the tetrahedron that contains the points 0,0,8 ,

0,6,0 , and 10,0,0 can be written as , ,x y z

Vw w w

with 0x . Determine the

sum x y z w .

17. Let p x and q x be quartic polynomial functions with integral coefficients such that

1 3p q , 2 4p q , 3 5p q , 4 6p q , and 5 7p q k , where k is a

positive multiple of 23. Determine the smallest value of k that satisfies these conditions.

18. In ABC with right angle at C , 56

65

AC

BA . tan BAC k . Determine the value of k .

Express your answer as an integer or as a common or improper fraction reduced to lowest terms.

19. 4

lim5

x

x

x ek

x

. Determine the value of k . Express your answer as an integer or a

common or improper fraction. 20. A plane flying at an average speed of 420 mph at an angle of 40 west of due north

encounters a wind pushing 20 mph at an angle of 20 west of due south. The plane will then be traveling at a new angle west of due north. Determine the degree measure of that angle. Express your answer as a decimal rounded to the nearest hundredth.


Pre-Calculus School(Use full school name – no abbreviations)



(Must be this reduced(“feet” or “ft.” optional. common fraction.)

1. 11.

(Must be this decimal,“feet” or “ft.” optional.)

2. 12.

("degrees" or

" "° optional.)

3. 13.

(Must be thisordered triple.)

4. 14.

(Must be this decimal.)

5. 15.

(Must be this decimal,“hours” or “hr.” optional.)

6. 16.

("degrees" or

" "° optional.)

7. 17.


8. 18.

(Must be this reduced(Must be this whole word.) common fraction.)

9. 19.

(Must be this reduced ("degrees" or

common fraction.) " "° optional.)

10. 20.

10

42.42

4

5

33

56

552

816

243

( )1,2,4

1320

3

13

27

8

9

Always

9−

115.3

1.7

1.848

4

1−

31.3

FROSH-SOPH EIGHT PERSON TEAM COMPETITIONICTM REGIONAL 2018 DIVISION AA PAGE 1 of 3

NO CALCULATORS


1. Determine the value of x so that 10 3 9 8x x− = − . Express your answer as an integer or as acommon or improper fraction.

2. a and b are integers so that ( ) ( )3 2 864a b = . Determine the value of ( )3 2a b .

3. Determine the numeric area of the right triangle marked as shown.

4. The greatest common factor of 4 977469x y and 743911x y is ( ) ( )w pkx y . Determine the value

of ( )k w p+ + .

5. Define 2 24 4a b a ab b⊗ = − + . Determine the value of ( )2 3 1− ⊗ ⊗ .

6. Consider the set { }11,17,10,11,22,11,14S = . The sum of the mode and the median for this set

is k larger than the arithmetic mean for this set. Determine the value of k . Express youranswer as an integer or as a common or improper fraction.

7. Two points are chosen on the circumference of a circle. Determine the probability that thechord connecting these two points is longer than the radius of the circle. Express youranswer as an integer or as a common fraction.

8. “The positive difference between the squares of two odd integers is divisible by n .”Determine the largest positive integer n such that this statement is necessarily true.

3

2x+1

x-1

1

2x+5


NO CALCULATORS


9. Ages are given in whole numbers of years. Ella is one year less than twice Aune’s age. In 7years, Aune will be the age that Ella is currently. Determine the age in whole number ofyears that Ella is currently.

10. In the diagram as shown, PQRS is a square and QAS is a quarter circle

with center at P . Point U is the midpoint of QR and T lies on RS

such that UT is tangent to QAS at A .TR k

UR w= where k and w are

relatively prime integers. Determine the sum ( )k w+ .

11. The line 2 4 0x y− + = is tangent to a certain circle at the point ( )0,2 . The line 2 7y x= − is

tangent to the same circle at the point ( )3, 1− . Determine the coordinates of the center ( ),h k

of this circle. Express your answer as the ordered pair ( ),h k .

12. (Multiple Choice) For your answer write the capital letter(s) that correspond to the bestanswer. (Note: k∼ means “not statement k .”)

Assume that the three conditional statements n w→ , t n→∼ , and c w→ ∼ are valid

(true). Determine which of the following conditional statement(s) must be valid.Express your answer as the capital letter(s) of those statements.

A) w t→∼ B) c t→ C) w c→∼ D) t w→∼ E) w n→

13. Consider the sequence 1 2 3 4 63S = + + + + + . Then 2k S− = . Determine the value of k .

14. Determine the sum of all integers for which the rational expression2

2

12

7 6

x x

x x

− −

− +has negative

values.

U

P

Q

S

R

AT


NO CALCULATORS


15. When written in simplest reduced form with a rationalized denominator,

2

2 3 5

a b c d f

g

+ +=

+ −where a , b , c , d , f , and g are integers with 0g > .

Determine the sum ( )a b c d f g+ + + + + .

16. 2 28! 29! 30!

28!n

+ += and 0n > . Determine the value of n .

17. Let w represent the width of a rectangle. The length of this rectangle is 5 less than 3 timesthe width. The values of w for which the width will be more than 5 times the length of the

rectangle is represented by k w p< < . Determine the sum ( )k p+ . Express your answer as

an integer or as a common or improper fraction.

18. Determine the value of k so that ( ) ( )38 32k k−= . Express your answer as an integer or as acommon or improper fraction.

19. Exactly 15 people are in a room. They all shake hands once with each of the other persons.Determine the total number of handshakes exchanged.

20. In the figure, O is between W and E , with50OW = and 10OE = . P , M , and O are

collinear. EP WM R= ∩ , EM WP Q=

∩ , and

RQ WE X= ∩ Determine the length of OX .

2018 RAA School ANSWERS

Fr/So 8 Person (Use full school name – no abbreviations)



(Must be this reduced (Must be thiscommon fraction.) ordered pair.)

1. 11.

(Must have all threecapital letters, any order.)

2. 12.

3. 13.

4. 14.

5. 15.


6. 16.

(Must be this reduced (Must be this reducedcommon fraction.) improper fraction.)

7. 17.


8. 18.

(“years old” or“years” optional.) (“handshakes” optional.)

9. 19.

10. 20.

2

3

25

105

15

2

145

42

30

7

15

8

2

3

( )1,0

A, B, D675

15

362

58

7

16

2018

32

2

JUNIOR-SENIOR EIGHT PERSON TEAM COMPETITIONICTM REGIONAL 2018 DIVISION AA PAGE 1 OF 3

NO CALCULATORS


1. Determine the exact sum ( )( )2018

5

2 2 1n

n n=

− −∑ .

2. The square of one more than the cube root of the value in cents of Wally’s pocket money isequal to Alaska’s rank in order of admission to the United States. Determine the number ofcents Wally has in his pocket. (Hint: Hawaii is our 50th state.)

3. Let log 5c a= and log 3c b= . Then 5log 405c

ka wb pc

q

+ += where k , w , p , and q are

relatively prime integers with 0q > . Determine the value of the sum ( )k w p q+ + + .

4. Let16

4lim

16x

xk

x→

−= −



5. Determine the value of k so that 3 28 2x x kx+ + − is divisible by ( )1x − . Express your


6. In the following representation, A and B are digits with 0A ≠ . Then a number x can beexpressed as ABBA and x is also the product of three consecutive prime numbers.Determine the sum of all possible values for x .

7. ( )f x and ( )g x are both linear functions. ( ) 2h x x k= + is a quadratic function, k a real

number. ( )( )( ) 23 5f g h x x= − and ( )( ) 3 1f g x x= + . Determine the value of k . Express



NO CALCULATORS


8. The graph of 2 10x y z+ + = is a plane. The graph of 6y z− = is another plane and these

two planes intersect in a line. This line intersects the plane 0x = at point ( ), ,x y z .

Determine the sum ( )x y z+ + . Express your answer as an integer or as a common or

improper fraction.

9. The graphs of 3y x= and 3y x= are plotted on the same set of axes. All points with x-

values of 2 and 1− are highlighted on these graphs. These highlighted points determine aconvex polygon. Determine the numerical area of this polygon.

10. Let1

8!M = and 1 1! 2 2! 3 3! 4 4! 9 9! 1N = + + + + + +i i i i i . Determine the product ( )MN .

11. w , x , y , and z are positive real numbers.

2

2

2 2

2 2 1200

wz wy zy

wz wx xz

wx wy xy

xz yz xy

+ = + =

+ = + + =

. Determine

the product ( )xyz .

12. ( ) ( )( )3 22 3 8 12x x x x k x w px q+ − − = + + + . Determine the sum ( )k w p q+ + + . Express


13. The arithmetic mean of nine different positive integers is 16. Determine the largest possiblevalue of the largest of these nine integers.

14. The area of the region that is inside the graph of 4sinr θ= but outside the graph 2r = can

be expressed ask w p

q

π +where k , w , and q are relatively prime integers with 0q > .

Determine the sum ( )k w p q+ + + .


NO CALCULATORS


15. The equation 3 23 6 12 0x x x+ − − = has solutions r , s , and t . The equation

3 2 0y ay by c+ + + = has solutions1

r,

1

s, and

1

t. Determine the sum ( )a b c+ + . Express


16. The sum of all x , 0 2x π≤ < , such that ( )2sin 3 1 0x − = can be expressed as kπ .


17. A single die is tossed until a “5” appears on the upper face. Determine the probability thatthe number of tosses was odd. Express your answer as an integer or as a common fraction.

18. Let 5 4 3 2 0x kx wx px qx c− + − + + = where 0k > , 0w > , 0p < , 0q < , and 0c < .

Determine the maximum number of possible positive roots for x in the given equation.

19. (Multiple Choice) For your answer write the capital letter(s) that corresponds to youranswer(s).

Determine which of the following six statements are true for all valid replacements of θ .

(A) sec cos 1θ θ = (B) sin cos2

π − θ = θ

(C) 1Sin csc− θ = θ

(D) 2 2sin cos 1θ− θ = − (E) 2

2

1tan 1

cosθ+ =

θ(F)

1sin 1

csc

θ =

θ

20. Consider the inequality2

02 4

x x

x x+ <

− −solved over the domain [ ]10,10− . Determine the

sum of all integers from this domain that are solutions for this inequality.


Jr/Sr 8 Person (Use full school name – no abbreviations)



1. 11.

(“cents” or“pennies” optional.)

2. 12.

3. 13.


4. 14.


5. 15.

6. 16.


7. 17.


8. 18.

(Must be these three capitalletters in any order.)

9. 19.

10. 20.

2014

216

90

3

26

3

2−

1001

7−

1

8

1

A, B, E

3 OR three

6

11

5

1

6

16

10811

5

4000

CALCULATING TEAM COMPETITION ICTM REGIONAL 2018 DIVISION AA PAGE 1 of 3 Round answers to four significant digits and write in standard notation unless otherwise specified in the question. Except where noted, angles are in radians. No units of measurement are required. 1. ABC is isosceles with base AC of length 14 and numeric area of 77. Determine the

degree measure of B . 2. One light year is 125.879 10 miles. (A light year is the distance light travels in one Julian

Earth year of 365.25 days.) The August 21, 2017 total eclipse of the sun lasted 2 minutes and 43 seconds. Determine the distance, in miles, light traveled during that eclipse. Express your answer in scientific notation.

3. The equation of the parabola which passes through points 1.6,16.18 , 3.2,43.22 , and

2.81, 2.018 can be written in the form 2y ax bx c . Determine the sum a b c .

4. Determine the smaller of the roots for x when 2log 20 18 2 log 3 5x x .

5. The North American Wood Chuck can chuck 3.784 uniformly sized planks of wood every

18 minutes. The South American Wood Chuck can chuck 2.282 of the same sized planks every 12 minutes. Determine how much more wood the North American Wood Chuck can chuck, measured in planks, in one day.

6. Let 1 1u and 2

1 12 4n n nu u u . Determine the value of 6u . Express your answer as

an exact integer. 7. The first Super Bowl 30-second advertisement sold for $42,000. For Super Bowl LI (51), a

30-second ad sold for $5,000,000. Assume the rate for Super Bowl ads increase at a constant annual rate compounded yearly, %k . Determine the value of k . Express your answer as that value without the percent sign.

CALCULATING TEAM COMPETITION ICTM REGIONAL 2018 DIVISION AA PAGE 2 of 3 8. A regular pentagon is inscribed in a circle with radius 20. Determine the numeric area of this

pentagon. Express your answer as a decimal rounded to the nearest thousandth. 9. Bonnie had $58.89 to spend. She bought 3 oranges at 47 cents apiece, 4 apples at 43cents

apiece, and four 6-bottle cartons of soda at $4.99 per 6-bottle carton. Determine the integral number of cents Bonnie had left from her $58.89 . (Ignore any taxes.)

10.

1 1

2 0 1 8 0 23

1 7 0 7 2 0

4 2

A

. Determine the sum of the entries in matrix A . Express your


11. In the diagram (not drawn to scale), chords AD and EH intersect the inner of concentric circles centered at P as shown.

4AB , 3EF , 41FG , and EH is 0.5 distance from center

P . Determine the distance of AD from center P . 12. Let 90 180x and sin( ) 0.6258x . Determine x , correct to the nearest second.

Express your answer in the form ' "k w p . 13. On flat land, a house is constructed with vertical walls that are 12.25 feet high. A ladder is

to be situated 5.469 feet from the base of a wall of the house. Determine the minimum number of feet in the length of this ladder so that the ladder can reach the top of the wall of the house.

C

B

G

F

P

A

D

E

H

CALCULATING TEAM COMPETITION ICTM REGIONAL 2018 DIVISION AA PAGE 3 of 3

14. 35 8x when x k . Determine the value of k .

15. Determine the minimum value for 3 25 16 15f x x x x over the domain interval

4,7 .

16. An equilateral triangle is inscribed in a circle. The sides of the triangle are of length 10.

Determine the numeric area inside the circle but exterior to the triangle. 17. A game is played with a special fair cubical die that has one red side, two blue sides, and

three green sides. The result of a roll is the color of the top side (facing up). The die is rolled repeatedly and the result recorded. Determine the probability that the third blue result occurs on the twelfth roll.

18. Determine the number of inches in the circumference of a circle whose area is 8946 square

inches. 19. A rectangle is inscribed in a circle with numeric area 37 . Determine the perimeter of the

largest such rectangle.

20. The function

42

281

9

x

x

ee

f xe

e

has a missing point, called a hole or point discontinuity.

Determine the coordinates of that hole. Express your answer as an ordered pair ,x y .


Calculator Team (Use full school name – no abbreviations)

Correct X 5 pts. ea. = Note: All answers must be written legibly. Round answers to four significant digits and

write in standard notation unless otherwise specified in the question. Except where noted, angles are in radians. No units of measurement are required.

("degrees" or " " optional.) (Accept 14.5 .) 1. 11. (Must be in scientific notation, “miles” or (Must be this answer “mi.” optional.) in this format.) 2. 12. (“feet” or “ft.” optional.) 3. 13. 4. 14. (“planks” optional.) 5. 15. (Must be this integer, comma use optional.) 6. 16. (Must be this decimal only with no “%” sign.) 7. 17. (“inches” or (Must be this decimal.) “in.” optional.) 8. 18. (Must be this integer, “cents” optional.) 9. 19. (Must be this integer.) (Must be this ordered pair.) 10. 20.

64.94

73.037 10

114.1

1.464

28.88

2090919

10.03

951.057

3580

336 0.1972,1.642

34.41

335.3

0.05299 OR .05299

61.42

67.13

2.322

13.42

141 15'33"

14.50

ICTM Math Contest

Freshman – Sophomore

2 Person Team

Division AA

FROSH-SOPH 2 PERSON COMPETITION LARGE PRINT QUESTION 1 ICTM 2018 REGIONAL DIVISION AA NO CALCULATORS ALLOWED

1. Let

2

2

3 2 4 3 2 1x x x

ax bx c

Determine the ordered triple , ,a b c . Express your answer as that ordered triple.


2. Point A has coordinates 2,a and lies on the line

3 2y x . Point B has coordinates ,2b and lies on the line 2 3 18x y . Determine the length of AB.


3. 8 is the arithmetic mean between 3 and x. 12 is the geometric mean between y and 36. Determine the sum x y .


4. If one of the composite numbers between 10 and 35, inclusive, is selected at random, determine the probability that the sum of the digits is an even number. Express your answer as an integer or as a common fraction.


5. A rectangle has sides with integral lengths and an area of 15. One side has length 2 1a and an adjacent side has length 3 1c . Determine the largest possible product ac . Express your answer as an integer or as a common or improper fraction.

FROSH-SOPH 2 PERSON COMPETITION LARGE PRINT QUESTION 6 ICTM 2018 REGIONAL DIVISION AA CALCULATORS ALLOWED

6. Marie’s average after 4 quizzes was 84. If she averages 90 on her next N quizzes, her overall average will be 87.6. When 22 5 12 0x x , the sum of the solutions for x is S and the product of the solutions for x is P. Determine the value 2 3N S P .


7. Point P ,x y is 0.25 of the way from 2,14 to

18,6 .

Let n be the number of integer values of k for which 2 2 12 10k .

Determine the sum x y n .


8. Set A contains all positive integral factors of 48 and set B contains all positive integral factors of 60. Determine the positive difference in the number of elements in A B and A B .


9. Nina can complete 2

3 of a

project in half a day. Let N represent the number of days it will take her to complete 12 of these projects. A rectangle has a side of 12 and a perimeter of 56. A quadrilateral with perimeter P is formed by connecting the midpoints of the sides of this rectangle. Determine the sum N P .


D

C B

A

10. Let

23 22 7 3 1R In right triangle ABC with CD AB ,

7AD , 9BD and CD W .

Determine the product RW .

FROSH-SOPH 2 PERSON COMPETITION EXTRA LARGE PRINT QUESTION 11 ICTM 2018 REGIONAL DIVISION AA CALCULATORS ALLOWED

11. The longer diagonal of a rhombus with an interior angle measuring 120 is 22 3. Let k represent the numeric perimeter of the rhombus. Let w represent the length of the shorter diagonal of the rhombus. Determine the value of k w .

FROSH-SOPH 2 PERSON COMPETITION EXTRA LARGE PRINT QUESTION 12 ICTM 2018 REGIONAL DIVISION AA CALCULATORS ALLOWED

12. The ratio of the degree measure of an interior angle to the degree measure of an exterior angle in a regular polygon is 10 :1. Determine the number of sides in that regular polygon.

FROSH-SOPH 2 PERSON COMPETITIONICTM 2018 REGIONAL DIVISION AA PAGE 1 OF 2

D

C B

A

1. Let ( ) ( )( )2 23 2 4 3 2 1x x x ax bx c− + − + = + + . Determine the ordered triple ( ), ,a b c .

Express your answer as that ordered triple.

2. Point A has coordinates ( )2,a and lies on the line 3 2y x= + . Point B has coordinates

( ),2b and lies on the line 2 3 18x y+ = . Determine the length of AB .

3. 8 is the arithmetic mean between 3 and x . 12 is the geometric mean between y and 36.

Determine the sum ( )x y+ .

4. If one of the composite numbers between 10 and 35, inclusive, is selected at random,determine the probability that the sum of the digits is an even number. Express your answeras an integer or as a common fraction.

5. A rectangle has sides with integral lengths and an area of 15. One side has length ( )2 1a +

and an adjacent side has length ( )3 1c − . Determine the largest possible product ( )ac .

Express your answer as an integer or as a common or improper fraction.

6. Marie’s average after 4 quizzes was 84. If she averages 90 on her next N quizzes, her

overall average will be 87.6. When 22 5 12 0x x− − = , the sum of the solutions for x is S

and the product of the solutions for x is P . Determine the value ( )2 3N S P+ + .

7. Point P ( ),x y is 0.25 of the way from ( )2,14− to ( )18,6 . Let n be the number of integer

values of k for which 2 2 12 10k− ≤ − < . Determine the sum ( )x y n+ + .

8. Set A contains all positive integral factors of 48 and set B contains all positive integralfactors of 60. Determine the positive difference in the number of elements in A B∪ and

A B∩ .

9. Nina can complete2

3of a project in half a day. Let N represent the number of days it will

take her to complete 12 of these projects. A rectangle has a side of 12 and a perimeter of 56.A quadrilateral with perimeter P is formed by connecting the midpoints of the sides of this

rectangle. Determine the sum ( )N P+ .

10. Let 23 22 7 3 1R = + + + + . In right triangle ABC

with CD AB⊥ , 7AD = , 9BD = and CD W= . Determine

the product ( )RW .

FROSH-SOPH 2 PERSON COMPETITION EXTRA QUESTIONS 11-15ICTM 2018 REGIONAL DIVISION AA

11. The longer diagonal of a rhombus with an interior angle measuring 120° is 22 3 . Let k

represent the numeric perimeter of the rhombus. Let w represent the length of the shorter

diagonal of the rhombus. Determine the value of ( )k w+ .

12. The ratio of the degree measure of an interior angle to the degree measure of an exterior angle in aregular polygon is 10 :1. Determine the number of sides in that regular polygon.


Fr/So 2 Person Team (Use full school name – no abbreviations)




(Must be this ordered triple.)1.

2.

3.(Must be this reduced common fraction.)

4.(Must be this reduced improper fraction.)

5.

6.

7.(“elements” optional.)

8.

9.

10.


Extra Questions:

11.(“sides” optional.)

12.

13.

14.

15.

* Scoring rules:





( )17, 14,1−

2 13

178

1914

3

7−

21

10

49

42

110

22

(Intentionally blank).




ICTM Math Contest

Junior – Senior

2 Person Team

Division AA

JUNIOR-SENIOR 2 PERSON COMPETITION LARGE PRINT QUESTION 1 ICTM 2018 REGIONAL DIVISION AA NO CALCULATORS ALLOWED

1. Let 1i . Determine the value of

220188 2 3 4i i .


2. Let represent the largest angle, 0 360 , such that 22 1 sin 1 cot sin

In pentagon PENTA, P E , A N , and

45T P N . Let T k .

Determine the value of sin k .


3. Determine the distance between the vertex of the graph of

23 12 4y x x and the center of the circle

2 2 6 8 50 0x y x y .


4. Determine the maximum value of the function , 5 2f x y x y within the region defined by 1 7x , 2 6y and

2 16x y .


5. Let 4 2f x x and

2 6g x x . Determine the value of the sum

12 2f g g f .

JUNIOR-SENIOR 2 PERSON COMPETITION LARGE PRINT QUESTION 6 ICTM 2018 REGIONAL DIVISION AA CALCULATORS ALLOWED

6. A bag contains 3 red marbles, 5 blue marbles, 4 white marbles and 2 green marbles. Two marbles are selected at random without replacement. Determine the probability that both marbles will be the same color. Express your answer as an integer or as a common fraction.


7. An arithmetic sequence and a geometric sequence each have first term 32 and fifth term 162. Determine the absolute value of the difference between the third terms of these two sequences.


8. Let w represent an odd positive integer and k represent an even positive integer such that

17w k . Determine the sum of all possible values of w.


E

D

C B

A

9. In ABC with right angle C,

1009BD , 2221AE ,

CE EB , and 2AD CD . Determine

the length of AB.


10. The number of bacteria in a sample is increasing at a rate of 8% per hour. Let h represent the number of hours until the number of bacteria doubles. Let s represent the sum of the values of x that satisfy 7 2 12 19x . Determine the sum h s . Express your answer as a decimal rounded to 4 significant digits.

JUNIOR-SENIOR 2 PERSON COMPETITION EXTRA LARGE PRINT QUESTION 11 ICTM 2018 REGIONAL DIVISION AA CALCULATORS ALLOWED

11. Let 2 0

1 8A

and

2 1

5 4B

. Determine

the value of the determinant of A B .

JUNIOR-SENIOR 2 PERSON COMPETITION EXTRA LARGE PRINT QUESTION 12 ICTM 2018 REGIONAL DIVISION AA CALCULATORS ALLOWED

12. Let 1i . The sum of the infinite series

2 327 9 3i i i can be written as x y i where x and y are real numbers. Determine the value of x y . Express your answer as a decimal.

JUNIOR-SENIOR 2 PERSON COMPETITIONICTM 2018 REGIONAL DIVISION AA PAGE 1 OF 2

E

D

C B

A

1. Let 1i = − . Determine the value of ( ) ( )220188 2 3 4i i+ + + .

2. Let θ represent the largest angle, 0 360° ≤ θ < ° , such that ( )22 1 sin 1 cot sin− θ − = θ θ . In

pentagon PENTA , P E∠ ≅ ∠ , A N∠ ≅ ∠ , and 45T P N∠ = ∠ +∠ − ° . Let T k∠ = ° .

Determine the value of ( )sin kθ+ .

3. Determine the distance between the vertex of the graph of 23 12 4y x x= − − − and the center

of the circle 2 2 6 8 50 0x y x y+ − + − = .

4. Determine the maximum value of the function ( ), 5 2f x y x y= − within the region defined

by 1 7x≤ ≤ , 2 6y≤ ≤ and 2 16x y+ ≤ .

5. Let ( ) 4 2f x x= − and ( ) 2 6g x x= − . Determine the value of the sum

( )( ) ( )( )12 2f g g f− + .

6. A bag contains 3 red marbles, 5 blue marbles, 4 white marbles and 2 green marbles. Twomarbles are selected at random without replacement. Determine the probability that bothmarbles will be the same color. Express your answer as an integer or as a common fraction.

7. An arithmetic sequence and a geometric sequence each have first term 32 and fifth term 162.Determine the absolute value of the difference between the third terms of these twosequences.

8. Let w represent an odd positive integer and k represent an even positive integer such that

17w k= − . Determine the sum of all possible values of w .

9. In ABC∆ with right angle C , 1009BD = , 2221AE = , CE EB= ,

and ( )2AD CD= . Determine the length of AB .

10. The number of bacteria in a sample is increasing at a rate of 8% per hour. Let h representthe number of hours until the number of bacteria doubles. Let s represent the sum of the

values of x that satisfy 7 2 12 19x − − = . Determine the sum ( )h s+ . Express your answer

as a decimal rounded to 4 significant digits.

JUNIOR-SENIOR 2 PERSON COMPETITIONICTM 2018 REGIONAL DIVISION AA PAGE 2 OF 2

11. Let2 0

1 8A

=

and2 1

5 4B

=

. Determine the value of the determinant of A B× .

12. Let 1i = − . The sum of the infinite series 2 327 9 3i i i− + − + ⋅⋅⋅ can be written as x y i+

where x and y are real numbers. Determine the value of ( )x y+ . Express your answer as a

decimal.


Jr/Sr 2 Person Team (Use full school name – no abbreviations)




1.

2.

3.

4.

5.(Must be this common fraction.)

6.

7.

8.

9.(Must be this decimal.)

10.


Extra Questions:

11.(Must be this decimal.)

12.

13.

14.

15.

* Scoring rules:





1 24i+ OR 24 1i +1

2OR 0.5 OR .5

13

31

4−20

91

25

680

53

9.578

48

16.2





III

II

IV

I

ORAL COMPETITION ICTM REGIONAL 2018 DIVISION AA 1. A scientific experiment is done on a particle and it is found that in the setup below the particle

will remain in one of the four areas for exactly one minute before moving through one of the circles into an adjacent area. The particle is equally likely (same probability) to move through any circle located in the area it is currently in to an adjacent area. No particle will remain in the area it is currently in during consecutive minutes. After the 10th minute, how much more likely is the particle to be in area I if it starts in area I than if it starts in area IV? Give your answer as a percent, rounded to the nearest tenth of a percent.

2. In a certain school cafeteria all students buy either cookies, brownies, or apples for lunch. For

unknown reasons, the head chef wants 30% of the students to buy cookies for lunch, 30% to buy brownies, and 40% to buy apples each day. She notices that of the students who buy an apple one day, 10% buy a cookie the next day and 20% buy a brownie. Also, of the students who buy a brownie one day, 30% buy a cookie the next day and nobody buys an apple. What percentage of students who buy a cookie one day need to switch to buying an apple the next day to maintain the desired percentages?

3. Let T be a 2x2 transition matrix with entries aij. Let X be a 2x1 steady state distribution vector

with entries bij such that TX=X and a12 = b11 = n. Prove that at least one of the following must be true: a11 = n or a12 = 0. (Note that aij and bij describe the entries in the ith row and the jth column of their respective matrix)

ORAL COMPETITION ICTM REGIONAL 2018 DIVISION AA EXTEMPORANEOUS QUESTIONS Give this sheet to the students at the beginning of the extemporaneous question period. STUDENTS: You will have a maximum of 3 minutes TOTAL to solve and present your solution to these problems. Either or both the presenter and the oral assistant may present the solutions.

1. Given the transition matrix T .3 .1 .2

.2 .6 .4

.5 .3 .4

A B C

A

B

C

, explain what the entry in row 3, column 2

represents. 2. Determine if each of the following could be a transition matrix:

a. 0.1 0.9

0.8 0.2

b. 0.3 1

0.7 0

c. 0.1 1.1

1.1 0.1

d. 0.2 0.5 0.3

0.8 0.5 0.7

3. Describe the process for determining if the matrix .7 1

T.3 0

is regular.

4. Given the regular stochastic transition matrix T = .6 .1

.4 .9

A B

A

B

, determine the steady state

distribution vector.

I.C.T.M. 2018 Regional Algebra I – Divisions 3AA, 4AA

I.C.T.M. 2018 Regional Algebra I – Divisions 3AA, 4AA 1. 3x -8 = 2+ 2x Þ x =10. 2. 50k = 4900 Þ k = 98. 3. y = (2 /3)x+ 2Þ 3x - 2y = -6. Report -6.

4. 47 y = 3+ 23x Þ y =

3+ 23x

47. Table gives (x, y)= (6,3). Report 279.

5. Values are x, 1.6x, (5/6)(1.6x), (.85)(5/6)(1.6x) = 272. Report x = 240. 6. Square to get x+ 16x =12, from which x2 - 40x+144 = 0. Roots are 4 and 36,

but only 4 checks. 7. 0 = (x - s)(x - s- 2)= x2 - (2s+ 2)x + s2 + 2s = 0. This gives s2 + 2s = 24, from

which s = 4, -6, and k = 2s+ 2 =10, -10. Report 10. 8. 25=16a+ 4k -3Þ112a+ 28k =196. 88= 49a+7k -3Þ196a+ 28k = 364.

Subtract to get the pair (2, -1). 9. The red-yellow-blue triples are 211, 311, 411, 511, 611, 322, 422, 522, 622, 433,

533, 633, 544, 644, 655. There are 15 of these so the probability is 15/216 = 5/72. Report 5/72.

10. 6x2 +10x+5=1Þ (3x+ 2)(x+1) = 0. Report both -1 and -2/3. 11. Let the original number be

htu =100h+10t +u =100h+10(u+ 4)+u =100h+11u+ 40. The second number is

100u+10t + h =100u+10(u+ 4)+ h =110u+ h+ 40. Subtract this from the original to get 99h-99u =198, or h = u+ 2. The possible original numbers are thus 351, 462, 573, 684, 795. Report the sum 2865

12. The side of the square is 6. The rectangle has sides 9 and 6. Report the area 54.

13.

G

R=

R

EÞ R2 =GE =160000 Þ R = 400. Report the total 800+ 200+ 400 =1400.

14. r = (5/3)w= (5/3)(w-182)Þ (5/3)w+ w+ w-182 = 2018Þ w= 600, so r =

1000. 15. Let x be the shorter leg. Then (2x -3)2 = x2 + (2x -11)2. Then x = 28 and the area

is (1/ 2)(28)(45)= 630.

I.C.T.M. 2018 Regional Algebra I – Divisions 3AA, 4AA

16. x+1 = 2 x+1- 2Þ x = 3, y =11. Report 8. 17. 40 = 25a+5b+ c, 32 = 9a -3b+ c Þ1= 2a+ b.

k = 9a+3b+ c, 3a+1= 4a+ 2b+ c = 2+ c. Then c = 3a -1 and k = 6a+ 2 k = 6a+ 2. Then 40 = 25a+5(1- 2a)+ (3a -1)Þ a = 2, b = -3, c = 5. Thus

k =18-9+5=14. 19. Simplify to get x < 6+ k. Since the largest integer solution is 18, k must be 13.

Report 13. 20. The equation is x3 -5x2 -9x+ 45= 0. Either use trial and error, hoping for

integer solutions, and get 5, 3 and -3. Or solve on your calculator. The possible equations for y, using 1 for a, are (y -5)(y -3) = y2 -8y + 24 = 0,

(y -5)(y+3) = y2 - 2 y -15= 0, and (y -3)(y+3) = y2 +0 y -9 = 0. The probability that b < 0 is thus 2/3. Report 2/3.

I.C.T.M. 2018 Regional Geometry – Divisions 3AA, 4AA

I.C.T.M. 2018 Regional Geometry – Divisions 3AA, 4AA 1. Draw to see that AT = 5, so y = 7, and HC =10, so x = 3. Report (3, 7).

2. From

3.2

h=

h

14.45 we get h = 6.8. The area is

(6.8)(17.65)

2= 60.01. Report 60.01.

3. Let the parallel to BC through D meet AB at E. This creates isosceles triangle

ADE with AD = AE = 9 and EB = 6, making AB =15. Report 15. Alternatively, if the problem is solvable, then the measure of angle B does not matter, so take it to be 45. The rest is easy.

4. Let y = (C - 2x) /3. Substitute to get

This has one solution for x, i.e., tangency, if

Solve to get C = -4±10 13. With 1 as

the denominator choose the plus sign and report -4+10+13+1= 20. 5. The acute angles in triangle CDE are both 15, so report 75. 6. Let AC = 5x and ED = 2x =CD. Let BC = 2y. Then DB = 2y - 2x. From the

similar triangles ABC and EBD we get

5x

2y=

2x

2y - 2x, from which

y

x=

5

3. Then

AC

BC=

5

2·

3

5=

3

2. Report k = 3.

7. Let RB = x and ST = y. Then 4x = 3y and y = (4 /3)x. The area ratio

ARE

TSE is

thus Let the area of triangle EST be 16w. Then the area of

triangle ERA is 81w. Let the altitude to ST of triangle EST be z. Then the altitude to AB of triangle ERA is (9 /4)z and the altitude to DT of triangle ADT

is (13/4)z. Since S is the midpoint of DT , the area ratio

ADT

EST is

(1/2)(13z / 4)(2ST )

(1/2)z(ST )=

13

2, so the area of triangle ADT is

13

2(16w) =104w.

Subtract to get the area of quadrilateral AESD to be 88w. The area of triangle RET is 52w-16w= 36w. The required ratio is thus 88w/36w = 22/9. Report 22/9.


8. Let the segments of the hypotenuse be x and y. The segments of the legs are x+ 4

and y+ 4. Then x+ 4( )2 + y+ 4( )2= x+ y( )2

, from which xy = 4(x+ y+ 4). And

Perimeter

Area=

2(x + y+ 4)

(x+ 4)(y + 4) /2=

4(x + y+ 4)

xy+ 4(x+ y)+16=

4(x+ y+ 4)

4(x+ y+ 4)+ 4(x+ y+ 4)=

1

2.

Report 1:2.

9. CE = ED = 6, so AE ^ ED. Thus AD = 62 + 42 = 2 13. Report 2 13. 10. AE = 4. Let the center be (p, q). Let E = (0,0), D = (6,0), A = (0,4). Then

(6- p)2 + q2 = r 2 = (-2- p)2 + q2 . Expand and simplify to give p = 2. Then q = 1/2 and r2 = 65/ 4 = k. Report 65/4.

11. Choose a coordinate system with the leftmost common vertex O being (0, 0) and

the rightmost being (4, 0). Then the upper left corner of the hexagon is A(1, 3). Let the top vertex of the square be M (2,2). Let OM and the top edge of the

hexagon meet at E( 3, 3). Then the area of triangle OAE is

(1/2)( 3 -1)( 3)=

3- 3

2. There are four such congruent triangles, whose total

area is 4(3- 3) /2. The area of the hexagon is 6·

22 34

. Divide this by the

previous value to get .244 and report 24.4. 12. Jake went 7 kilometers east and 24 north, ending up 25 kilometers from his

starting point, the last part taking 3/2 of an hour. He thus traveled 56 kilometers in 4 hours for an average speed if 14 kph. Repot 14.

13. The three small right triangles have areas 4, 4 and 1. OLG has area 18. This

leaves 9 for the area of MATH. Report 9. 14. Let the circle have equation (x - a)2 + (y - b)2 = r 2. Substitute and get

a2 +36-12b+ b2 = r 2 , 9-6a+ a2 + 49-14b+ b2 = r2, and

49-14a+ a2 + 25-10b+ b2 = r2. These yield 22-6a- 2b = 0 and 38-14a+ 2b= 0. Add to get a = 3 and b = 2. Then r 2 = 25= k. Report 25.

15. Let s be the side of the square. Then the side of the triangle is s / cos15. The

double-angle formula gives cos30 = 2cos2 15-1, so cos2 15=

3 /2+1

2=

3+ 2

4.

The area of the triangle is

s2

cos2 15

3

4=

s2

( 3+ 2) /4

3

4=

s2 3

3+ 2. Rationalize the

denominator to give 2 3 -3 = k. Report 2 3 -3.


16. HOLDLet x be the number of minutes after 1:30 that Xavier arrives and let y be the same for Evie. If Evie arrives first, then y - x £15. If Xavier arrives first, then x - y £ 7. These are the conditions for a meet. Graph the lines y = x+15 and y = x -7 and the square with corners (0, 0), (75, 0), (75, 75) and (0, 75). The hexagon has area 752 -602 /2-682 /2. Divide this by 752, the area of the square, to get the probability of meeting 1513/5625. Report 1513/5625.

17. Let z be the tangent lengths from A. Then YC +6- z = 7 and DX + z = 9. This

makes YC + DX +6- z + z = 7+9 =16, so YC + DX =10. Report 10. 18. Let BH be the altitude from B, which is 30. Then triangle BDH is an 8-15-17

right triangle and DH = 16. Then DC = 24 and AD = 64+900 = 2 241. The perimeter is thus 4 241+32. Report 4 241+32.

19. The areas of ABCD, ABP and BQC are 8, 2 and 2, so the area of BPDQ is 4.

Report 4. 20. Let x be the distance between chords and let y be the distance from the center to

AB. Then r2 = y2 +102, r 2 = ( y+ x)2 +82, r 2 = (y + 2x)2 + 42. Simplify to get

2xy+ x2 = 36, 4xy+ 4x2 = 84. Then x2 = 6 and y =

30

2 6 and

r2 =

225

6+100 =

825

6. Thus

r =

5 22

2. Report 5 + 22 + 2 = 29.

I.C.T.M. 2018 Regional Algebra II – Divisions 3AA, 4AA

I.C.T.M. 2018 Regional Algebra II – Divisions 3AA, 4AA 1. x1/3 =1/3Þ x =1/27 Þ x-1 = 27 2. log(x2 -14x +5) = 2Þ x2 -14x+5=100Þ x2 -14x -95= 0Þ x =19, -5. Report

14.

3. 70d = -210Þ d = -3Þ a1 =14. The required sum is

(14-3n) =n=0

776

å 14-3n=0

776

å nn=0

776

å

=14·777 -3

776·777

2= -893550.

4. (x+ y)2 = x2 + y2 + 2xy Þ 36 =10+ 2xy.

(x - y)2 = x2 + y2 - 2xy =10- 2xy =10- 26 = -16. Report -16.

5. 20 =

12

1- rÞ r = 2 /5Þ a3 = (12)(2 /5)2 = 48/25.

6. The parabolas y = x2 +3 and y = 3- x2 meet when x =± 3, so [a,b]= [- 3, 3].

When x = 0, y = 3, 9, so [c,d]= [3,9]. Report 12. 7. Use (0, 1/3) from the first line and use the point-line distance formula to get

d =

2·0+ (-3)(1/3)+5

22 +32=

4 13

13. Report 30.

8. log 45= log5+ log9, so 2d = log9 = 2log3, so a2 = log5+ log3= log15, so k = 15. 9. The perpendicular bisector has slope 1/3 and passes through (4, 4). One equation

is y - 4 = (1/3)(x - 4), which converts to 1x -3y+8 = 0. Report 6.

10. y =

2x+1

3Þ x =

3y -1

2Þ f -1(x)=

3x -1

2. Report 4.

11. x =

64zw3

y=

64x2x6

x4 = 64x4 Þ x3 =1

64Þ x =

1

4.

12.

y =

4x-

1x+ 23x2

· x2 (x+ 2)=4x(x + 2)- x2

3(x+ 2)=

3x2 +8x

3x+6. Report 20.

13. a = -5 and c = -6, so f (x)= x3 -5x2 + bx -6, so 0 = 8- 20+ 2b-6Þ b= 9.

I.C.T.M. 2018 Regional Algebra II – Divisions 3AA, 4AA

14. x+ x2 +1 = 8Þ x2 +1= (8- x)2 = 64-16x+ x2 Þ x = 63/16. 15. w= 2, k = -1, p = 3. Report k = 4. 16. d1 = 45t1, 2d1 = 81t2 . t2 = (90 /81)t1 = (10 /9)t1. t1+ t2 = (19 /9)t1. Average speed is

thus

135t1(19 /9)t1

=1215

19 mph.

17.

m+ n

mn= -4 and

m3 + n3

m3n3 = -316 =m+ n( )3 -3(m2n+mn2 )

m3n3 =m+ n( )3

m3n3 -3(m2n+mn2 )

m3n3 =

-64-

3mn(m+ n)

m3n3 = -64-3(m+ n)

m2n2 = -64-3·-4mn

m2n2 = -64+12

mn. Thus

-252 =

12

(m+ n) /(-4)=

-48

m+ n. Thus m+ n = 48/ 252 = 4 /21.

18. c

2 = a2 - b2 Þ 25= 81- b2 Þ b= 2 14. The length of the minor axis is thus

4 14. 19. a- b+ c = 0, 9a+3b+ c = 0Þ b = -2a, c = -3a. Since c = -9, report (3, -6, -9). 20. The solution region is the triangle with vertices (4, 2), (4, 1) and (10/3, 5/3). The

horizontal altitude is 2/3 and the corresponding vertical base is 1, so the area is 1/3.

I.C.T.M. 2018 Regional Pre-Calculus – Divisions 3AA, 4AA

I.C.T.M. 2018 Regional Pre-Calculus – Divisions 3AA, 4AA 1. Graph in polar mode to see that the path is a circle with radius 5/2, traversed twice

in the given q interval, so the distance traveled is 2·2p(5/2)=10p . Report 10. 2. d = 24 /cos40 = 31.3. 3. 3x+ 4y = 3 and 2x+3y = 4. Subtract to get x + y = -1 and report -1.

4. Geometric series with sum 8·

1/3

1-1/3= 4.

5. The points are (-1,0) and (1/ 2,1/ 2). Distance is (1/ 2 +1)2 +1/2 =1.848.

6. Law of Cosines gives 162 +142 - 2·16·14·1/2 t = 26.424. Then t = 1.7.

7. tanq =

m2 -m1

1+m2m1

=5-1/4

1+5/4=

19

9Þq = 64.7 or 115.3. Report 115.3.

8. The dot product is (-6)(117)+ (-78k )= 0, so k = -9. 9. Always 10. (sin x - cos x) =1/3Þ sin2 x - 2sin x cos x+ cos2 x =1/9 =1- 2sin x cos x

=1- sin2x, so sin2x = 8/9. 11. When A,B are chosen, the probability of the same color is

C(3,1)C(2,1)+C(6,1)C(1,1)

C(9,1)C(3,1)=

6+6

27=

12

27. When A,C are chosen, the probability

of the same color is

C(3,1)C(2,1)+C(6,1)C(4,1)

C(9,1)C(6,1)=

6+ 24

54=

15

27. When B,C are

chosen, the probability of the same color is

C(2,1)C(2,1)+C(1,1)C(4,1)

C(3,1)C(6,1)=

4+ 4

18=

12

27. Each of these must be multiplied by

1/3, the probability of the box choice, so the result is 13/27. 12. The amplitude is 3. 13. The angles are 60, 240, 420 and 600. Report 1320.

14.

1

21/3 -1·

22/3 + 21/3 +1

22/3 + 21/3 +1=

22/3 + 21/3 +1

2-1=

41/3 + 21/3 +1

1=1+ 21/3 + 41/3. Report (1,2,4).

I.C.T.M. 2018 Regional Pre-Calculus – Divisions 3AA, 4AA

15. Use the circle x2 + y2 =17 since it contains an easy point B(1, 4) on line b. The

circle meets the line a at the point A 2 221 /13, 3 221 /13( ), found from

substituting 2/3x for y in the equation of the circle. The midpoint M of AB is

. The slope of OM , which lies on the angle

bisecting line L, is

52+3 221

13+ 2 221, which reduces to

10+1 221

11. Report 243.

16. An equation of the plane is

x

10+

y

6+

z

8= 1, or 12x+ 20 y+15z =120. The vector

n = (12, 20,15) is perpendicular to the plane, so .

Report 816.

17. Let R(x) = p(x) – q(x + 2). Then R(1), R(2), R(3), R(4) are all 0, so

R(x)= a(x -1)(x - 2)(x -3)(x - 4) and R(5) = 24a, which must be a multiple of 23, the smallest being 24·23= 552. (Not my solution.)

18. The other leg is 33, so tan A= 33/56. 19. as , so the limit is 4/5. 20. Let p = (-420sin40, 420cos40) be the vector corresponding to the plane with no

wind. Let w= (-20sin20, - 20cos20) be the wind vector. The resultant vector representing the plane’s actual path is

r = (-420sin40- 20sin20, 420cos40- 20cos20) = r = (-276.81, 302.94). The

angle q for this vector is Report 42.42.

I.C.T.M. 2018 Regional Freshman-Sophomore 8-Person – Divisions 3AA, 4AA

I.C.T.M. 2018 Freshman-Sophomore 8-Person – Divisions 3AA, 4AA 1. 18x =12Þ x = 2 /3 2. 864 = 33 ·25 Þ a = 3, b= 5. Report 33 ·52 = 6755. 3. (1+3/2x)2 = (5+1/2x)2 + (x -1)2 Þ x = 5. The area is (1/ 2)(5+5/2)(5-1) =15. 4. 77469 = 3·72 ·17·31 and 43911= 32 ·7·17·41, so the greatest common factor

is 357x4 y1. Report 362. 5. -2Ä (3Ä1) = -2Ä (9-12+ 4)= -2Ä1= 4+8+ 4 =16 6. Mode = 11; median = 11; 96/7 = mean; k = 22 – 96/7 = 58/7. Report 58/7. 7. On the semi-circle the angle between the chords must be greater than 60. P = 2/3. 8. (2m+1)2 - (2k +1)2 = 4(m2 - k 2 )+ 4(m- k ) = 4(m- k)(m+ k +1). One of these

factors is even, the other odd, so 8 must divide the original difference. Report 8. 9. e = 2a -1, e = a+7. Thus a = 8 and e = 15. Report 15. 10. Let QU = RU = x and let RT = y. Then UA= x and ST = 2x - y. Also,

UT = 3x - y, 3x - y( )2

= x2 + y2 Þ 8x2 -6xy = 0Þy

x=

k

w=

4

3. Report 7.

11. The lines perpendicular to the tangents are y - 2 = -2(x -0)Þ y = -2x+ 2 and

y+1= -1/2(x -3)Þ y = -1/2x+1/2. These meet at (1, 0). Report (1, 0). 12. The contrapositives of the three given implications are and

. These are valid. The first two of these give so A is valid. The first of these and the third given give so B is valid. D is the contrapositive of A, which is valid, so D is valid. C is the converse of the first given and converses are not necessarily valid. Ditto for E. Report A, B, D.

13. S = 63·64 / 2 = 2016, so k = 2018.

14. Factor as

(x - 4)(x+3)

(x -6)(x -1). This is negative at -1, -2 and 5. Report the sum 2.

15.

2

2 + 3 - 5·

2 + 3+ 5

2 + 3+ 5=

2+ 6 + 10

2 + 3( )2-5

=2+ 6 + 10

2 6 =

3+1 6 +1 15

6.

Report 3+1+6+1+15+6 = 32.


16. n2 =

28!(1+ 29+ 29·30)

28!= 302. Report30.

17. L = 3W -5ÞW > 5/3. W > 5LÞW >15W - 25ÞW < 25/14. Then

70 /42 <W < 75/45. Report 145/42. 18. 23k = 25k-15 Þ k =15/2.

19. C(15,2) =

15·14

1·2=105

20. There is no restriction on ÐMOX , so make it 90 and choose coordinates so that

O = (0,0), E = (10,0), W = (-50,0), M = (0,50), P = (0,10). Lines WM and EP

have equations y = x +50 y = -x+10, giving R = (-20, 30). Lines EM and WP have equations y = -5x+50 and y = x /5+10, giving Q = (100 /13, 150 /13).

Then QR has slope

30-150 /13

-20-100 /13=

240

-360= -

2

3 and equation

y -30 = -

2

3(x+ 20).

Put y = 0 to get 45= k + 20, so k = 25. Footnote: Google “complete quadrangle” and “harmonic conjugate of a point

with respect to two others on their line.”

I.C.T.M. 2018 Regional Junior-Senior 8-Person – Divisions 3AA, 4AA


1.

1n=5

2018

å = 2018- 4 = 2014

2. w1/3 +1( )2

= 49Þ w1/3 = 6Þ w = 216

3. logc 4051/5 c1/5( ) = 1

5logc 405+

1

5logc =

1

5logc 34 ·5( ) + 1

5·1=

1

54logc 3+ logc 3+1[ ]

=

1a+ 4b+1

5. Report 1 + 4 + 1 + 5 = 11.

4.

5. Put x = 1 to get 0 =1+8+ k - 2Þ k = -7. 6. 7·11·13=1001 and this is the only set that works. Report 1001. 7. Let f (x)= ax + b and g(x)= cx+ d . Then f (g(h(x)))= f (g(x2 + k))= f (cx2 + ck + d) = acx2 + ack + ad + b = 3x2 -5.

Thus ac = 3 and ack + ad + b= -5. Also, f (g(x))= acx+ ad + b = 3x+1= 3x+ ad + bÞ ad + b =1. Then 3k +1= -5,

so k = -2. 8. Put x = 0 on both planes. The line of intersection meets the plane x = 0 at

(0, y0 , z0 ), where y0 + 2z0 =10 and y0 - z0 = 6. Thus z0 = 4 /3 and y0 = 22 /3. Report 0+ 22 /3+ 4 /3= 26 /3.

9. The polygon is a triangle with vertices (2, 8), (-1, 1) and (-1, -1). The area is 3. 10. Use (n+1)!= (n+1)(n!) = n(n!)+ n! , so n(n!) = (n+1)! - (n!) . Thus N = 2!-1!( ) + 3!-2!( ) + 4!-3!( ) + 5!-4!( ) + 6!-5!( ) + 7!-6!( ) + 8!-7!( ) + 9!-8!( ) + 10!-9!( ) +1 .

This reduces to 10!. Since M =1/8!, the product MN =10·9 = 90. 11. Subtract the second equation from the first to get w(y - x)= z(y - x), or

(w- z)(y - x)= 0. If w = z, then 2w2 = 0, which is given to be false. Thus y = x. Put y = x in the third equation to get 4wx = x2 Þ x = 4w. Put x = 4w in the second equation to get 2wz+ 4w2 = 4wz Þ z = 2w = x / 2Þ x = 2z = y. The fourth equation then gives 4z2 + 4z2 + 4z2 =1200 Þ z =10, x = y = 20. Thus xyz = 4000.

12. Factor to get 2x3 +3x2 -8x -12 = (x - 2)(x + 2)(2x+3). Report 2 – 2 + 2 + 3 = 5.


13. The sum of the nine positive integers is 144. Use 1 through 8, whose sum is 36. The largest is then 108.

14. r = 4sinq Þ r2 = 4r sinq, which converts to x

2 + y2 - 4y = 0 = x2 + (y - 2)2 = 4, showing the first graph to be a circle with center (0, 2) and radius 2. The second graph is a circle with center at (0, 0) and radius 2. These circles meet at (± 3, 1). The required region is the first circle minus two congruent circular segments, each

with area (1/3)p ·22 - (1/2)·2·2sin120 =

4p3- 3. The answer, then, is

Report 4 + 6 + 3 + 3 = 16.

15. -12 y3 -6y2 +3y+1= 0 has the reciprocal roots. Write this as

y3 +

1

2y2 -

1

4y -

1

12= 0. Report

1

2-

1

4-

1

12=

1

6.

16.

1

2= sin3x, so

3x =

p6

,5p6

,13p

6,17p

6,

25p6

,29p

6, and

x =

p18

,5p18

,13p18

,17p18

,25p18

,29p18

. The sum of these is 5p . Report 5.

17.

18. There are three sign changes. Decartes’ Rule of Signs gives 3 as the maximum

number of positive roots. Report 3. 19. A, B, E

20. The left side becomes

x2 +3x

(x - 2)(x+ 2), which is positive outside [-2, 2]. Of -1, 0

and 1, only 1 gives a negative value, so report 1.

I.C.T.M. 2018 Regional Calculating – Divisions 3AA, 4AA

I.C.T.M. 2018 Regional Calculating – Divisions 3AA, 4AA 1. The altitude is h = 7 / tan(B / 2) and the area is (1/ 2)(14)·7 / tan(B / 2) = 77. So

tan(B / 2) = 49 /77, from which B = 64.94.

2. miles.

3. Use Solve for a, b, c with “and” between these three equations to get

a = 81.897, b = -376.204,c = 408.451 and report 114.1. 16.18 = a(1.6)2 + b(1.6)+ c

43.22 = a(3.2)2 + b(3.2)+ c -2.018 = a(2.81)2 + b(2.81)+ c

4.

20x2 +18

3x+5=100. Solve gives -1.464 and 16.46. Report -1.464.

5. (3.784) 24·60 /18( ) - (2.282) 24·60 /12( ) = 302.72- 273.84 = 28.88. 6. The first six numbers are 1, 3, 7, 39, 1447, 2090919. Report 2090919.

7. 5000000 = 42000 1+ k /100( )50 Þ ln(5000 /42)= k · ln(1+ k /100) Þ k =10.03. 8. The total area of five triangles is 5· (1/2)(202 )sin72 = 951.057. 9. 58.89- 3(.47)+ 4(.43)+ 4(4.99)[ ] = 35.80. Report 3580.

10. The product of the matrices is , so the total is 3(112) = 336.

11. Let x be the distance from P to K on Let R and r be the large and small

radii. Then . Now

.

From these subtraction gives 16 + 8BK = R2 - r2 = 132, so Finally, x = 14.5.

12. The arcsin tool give x = 38.740929 degrees as an acute angle. Now and , which gives the obtuse

angle Report

13.

14.

I.C.T.M. 2018 Regional Calculating – Divisions 3AA, 4AA

15. The minimum value is -67.127 at x = 4.514668, so report -67.13.

16. The radius is , so the required area is

17. The probability of exactly two blues in the first 11 rolls is , so the probability that the third blue occurs on the twelfth roll is

18.

19. r = 37 and the side of the inscribed square is 74 and the perimeter is

20. The hole occurs where the denominator is 0, so solve to get

and thus Report


I.C.T.M. 2018 Regional Freshman-Sophomore 2-Person – Divisions 3AA, 4AA 1. Expand to get 17x2 -14x+1 and report (17, -14, 1). 2. A= (2,8) and B = (6,2). AB = 16+36 = 2 13. 3. x = 13 and y = 4. Report 17, 4. The composites are 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32,

33, 34, 35. There are 19 of these, of which 15, 20, 22, 24, 26, 28, 33, 35 have even digit sums. The probability is thus 8/19.

5. 2a+1= 5, 3c-1= 3Þ ac = 2(4 /3)= 8/3. 2a+1= 3, 3c-1= 5Þ ac =1·2 = 2.

2a+1=15, 3c-1=1Þ ac = 7·2 /3=14 /3. 2a+1=1, 3c-1=15Þ a = 0. Report 14/3.

6.

4·84+90N

4+ N= 87.6Þ N = 6. S = 5/2 and P = -6. N +2S +3P = -7.

7. P = (-2+ 20 /4, 14-8/4) = (3, 12)= (x, y). 5£ k <11Þ n = 6. Report 21. 8. A= {1,2,3,4,6,8,12,16,24,48}. B = {1,2,3,4,5,6,10,12,15,20,30,60}

AÇ B = {1,2,3,4,6,12}. n AÇ B( ) = 6 .

n(AÈ B)= n(A)+ n(B)- n AÇ B( ) =10+12-6 =16. Report 10. 9. 12 /(2 /3) =18 = number of half-days, so N = 9. The sides of the rectangle are 12

and 16 and the diagonal is 20. The perimeter P is thus 40. Report 49.

10. R = 23+ 22+ 7+ 3+ 1 = 23+ 22+ 9 = 23+ 25 = 2 7.

BD

CD=

CD

ADÞ

9

W=

W

7ÞW = 3 7. Then RW = 42.


I.C.T.M. 2018 Regional Junior-Senior 2-Person – Divisions 3AA, 4AA 1. i2018 = -1 so the first term is 8. (3+ 4i)2 = 9+ 24i-16, so report 1+24i. 2. Simplify to 2cos2q - cosq -1= 0 = (2cosq +1)(cosq -1)= 0, from which

cosq =1, -1/2. The largest value of q less than 360 is 240. With ÐP = x and

ÐA = y we get 3x+3y - 45= 540, so ÐT = (540+ 45) /3- 45=150 = k. Then

sin(q + k) = sin390 = sin30 =1/2. Report 1/2. 3. The parabola is y = -3(x+ 2)2 +8 with vertex (-2, 8). The circle is

(x -3)2 + (y+ 4)2 = 75 with center (3, -4). The distance is

(-2-3)2 + (8+ 4)2 =13. Report 13. 4. The region is a pentagon with one vertex at (7, 2), which has the largest x and the

smallest y, hence 5x - 2y is greatest, 31, there. Report 31. 5. f (g(2))= f (-2)= -10. g

-1( f (2))= g-1(6) = 6. Report -4.

6. P =

C(3,2)+C(5,2)+C(4,2)+C(2,2)

C(14,2)=

3+10+6+1

91=

20

91.

7. 162 = 32+ 4d Þ d = 65/ 2Þ a3 = 32+65= 97.

162 = 32r 4 Þ r =±3/2Þ g3 = 32(9/4) = 72. Report 25. 8. Solutions are (k, w)= (4,225), (16,169), (36,121), (64,81), (100,49),

(144,25), (196,9), (256,1). Report 680. 9. Let CD = x, CE = y. Triangle CDB gives x

2 + 4y2 =1009. Triangle ACE gives

9x2 + y2 = 2221. Multiply the first by 9 and subtract to give 35y2 = 6860, so

y =14 and x = 15. Then AB = 9x2 + 4y2 = 9·225+ 4·196 =53.

10. 2n = n(1+ .08)h Þ h =

ln2

ln1.08= 9.006468.

7x - 2 = -31, 31. Then Report 9.578.

III

II

IV

I

ORAL COMPETITION PAGE 1 OF 2 ICTM REGIONAL 2018 DIVISION AA JUDGES’ SOLUTIONS 1. A scientific experiment is done on a particle and it is found that in the setup below the particle

will remain in one of the four areas for exactly one minute before moving through one of the circles into an adjacent area. The particle is equally likely (same probability) to move through any circle located in the area it is currently in to an adjacent area. No particle will remain in the area it is currently in during consecutive minutes. After the 10th minute, how much more likely is the particle to be in area I if it starts in area I than if it starts in area IV? Give your answer as a percent, rounded to the nearest tenth of a percent.

SOLUTION: The transition matrix would be

𝑰 𝑰𝑰 𝑰𝑰𝑰 𝑰𝑽𝑰𝑰𝑰𝑰𝑰𝑰𝑰𝑽

𝟎 𝟎 𝟎 𝟎. 𝟓𝟎 𝟎 𝟎. 𝟓 𝟎. 𝟐𝟓𝟎𝟏

𝟎. 𝟓𝟎. 𝟓

𝟎 𝟎. 𝟐𝟓𝟎. 𝟓 𝟎

. The product to find

the distribution vector if the particle starts in area I is

𝟎 𝟎 𝟎 𝟎. 𝟓𝟎 𝟎 𝟎. 𝟓 𝟎. 𝟐𝟓𝟎𝟏

𝟎. 𝟓𝟎. 𝟓

𝟎 𝟎. 𝟐𝟓𝟎. 𝟓 𝟎

𝟏𝟎𝟏𝟎𝟎𝟎

, which

yields

𝟎. 𝟐𝟒𝟖𝟎. 𝟐𝟏𝟓𝟎. 𝟐𝟏𝟓𝟎. 𝟑𝟐𝟐

. The product if the vectors starts in area IV is

𝟎 𝟎 𝟎 𝟎. 𝟓𝟎 𝟎 𝟎. 𝟓 𝟎. 𝟐𝟓𝟎𝟏

𝟎. 𝟓𝟎. 𝟓

𝟎 𝟎. 𝟐𝟓𝟎. 𝟓 𝟎

𝟏𝟎𝟎𝟎𝟎𝟏

which yields

𝟎. 𝟏𝟔𝟏𝟎. 𝟏𝟖𝟖𝟎. 𝟏𝟖𝟖𝟎. 𝟒𝟔𝟑

. Subtracting 0.248 - 0.161 yields .087 or 8.7%

ORAL COMPETITION PAGE 2 OF 2 ICTM REGIONAL 2018 DIVISION AA JUDGES’ SOLUTIONS 2. In a certain school cafeteria all students buy either cookies, brownies, or apples for lunch. For

unknown reasons, the head chef wants 30% of the students to buy cookies for lunch, 30% to buy brownies, and 40% to buy apples each day. She notices that of the students who buy an apple one day, 10% buy a cookie the next day and 20% buy a brownie. Also, of the students who buy a brownie one day, 30% buy a cookie the next day and nobody buys an apple. What percentage of students who buy a cookie one day need to switch to buying an apple the next day to maintain the desired percentages?

SOLUTION: The transition matrix

C B A

C 0.3 0.1

B 0.7 0.2

A 0 0.7

x

y

z

and the desired steady state distribution

vector

C 0.3

B 0.3

A 0.4

describe the situation. An equation can be written and solved for z, the desired

outcome.

0.3 0.1 0.3 0.3

0.7 0.2 0.3 = 0.3

0 0.7 0.4 0.4

x

y

z

. Using the third row, this yields the equation 0.3z + 0*0.3 +

0.7*0.4 = 0.4, which when solved will give z = .4, so 40% of students need to switch from cookies to apples each day in order to achieve the desired results. 3. Let T be a 2x2 transition matrix with entries aij. Let X be a 2x1 steady state distribution vector

with entries bij such that TX=X and a12 = b11 = n. Prove that at least one of the following must be true: a11 = n or a12 = 0. (Note that aij and bij describe the entries in the ith row and the jth column of their respective matrix)

SOLUTION: The conditions give the equation

a n n n=

b c d d. As T is a transition matrix and

X is a steady state distribution vector, the entries in each column must add to 1. This yields

1b = - a , 1c = - n and 1d = - n . The equation can then be written as 1- 1- 1- 1-

a n n n=

a n n n.

Multiplying these matrices, using the first row of T with the first column of X gives the equation a*n + n*(1-n) = n. This can be rewritten as a*n – n2 = 0. Factoring, this becomes n(a – n) = 0; applying the zero product property gives n = 0 or n = a. Since a11=a, if n = a, then a11=n; Since a12=n, if n = 0, then a12=0.

ORAL COMPETITION PAGE 1 OF 2 ICTM REGIONAL 2018 DIVISION AA JUDGES’ SOLUTIONS EXTEMPORANEOUS QUESTIONS Give this sheet to the students at the beginning of the extemporaneous question period. STUDENTS: You will have a maximum of 3 minutes TOTAL to solve and present your solution to these problems. Either or both the presenter and the oral assistant may present the solutions.

1. Given the transition matrix T .3 .1 .2

.2 .6 .4

.5 .3 .4

A B C

A

B

C

, explain what the entry in row 3, column 2

represents. SOLUTION: 30% of those currently in state B will move to state C at the next stage. 2. Determine if each of the following could be a transition matrix:

a. 0.1 0.9

0.8 0.2

b. 0.3 1

0.7 0

c. 0.1 1.1

1.1 0.1

d. 0.2 0.5 0.3

0.8 0.5 0.7

SOLUTION: a) This cannot be a transition matrix as each of its columns does not each sum to 1. b) This is a valid transition matrix as the sum of each column is 1 and each element is between 0 and 1, inclusive. c) This is not a valid transition matrix because while the sum of each column is 1, the elements are not all between 0 and 1, inclusive. d) This is not a valid transition matrix as it is not a square matrix.

ORAL COMPETITION PAGE 1 OF 2 ICTM REGIONAL 2018 DIVISION AA JUDGES’ SOLUTIONS EXTEMPORANEOUS QUESTIONS

3. Describe the process for determining if the matrix .7 1

T.3 0

is regular.

SOLUTION: A regular matrix must be stochastic, which this is since it is square and the entries

in each column sum to 1, and must have some power of T with entries that are all positive. Since one of the entries in T is 0, look at powers of T to determine if there is a power where

all elements are positive. (Note that this happens with

2 0.79 0.7T =

0.21 0.3 . Students may

find this matrix, but do not need to – describing the process is sufficient)

4. Given the regular stochastic transition matrix T = .6 .1

.4 .9

A B

A

B

, determine the steady state

distribution vector.

SOLUTION: Let the steady-state distribution vector X =

x

y. Then

.6 .1=

.4 .9

x x

y y , which

when completing the multiplication, gives:

.6 + .1 =.4 - .1 = 0.

.4 + .9 =

x y xx y

x y y Since + = 1x y ,

this leads to the system

.4 .1 = 0 .4 .1 = 0= .1 = .2, = .8

+ = 1 1 + .1 = .1

x - y x - y.5x x y

x y . x y so

the steady state distribution vector is

0.2

0.8 or

1

54

5

WRITTEN AREA COMPETITION ALGEBRA I ICTM REGIONAL 2018 ...

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