WRITTEN AREA COMPETITION ALGEBRA I ICTM STATE 2010 DIVISION AA PAGE 1 OF 3 1. From the three equations 2 3x x= + , 5.12 3.4 18.76x + = , and 7 2 1x− = − , one equation is

selected at random. Find the probability that 3x = is a solution for the equation chosen. Express your answer as a common fraction reduced to lowest terms.

2. When 5x k+ is divided by ( )3x + , the remainder is known to be 31− . Find the value of

k . 3. 2 18k+ = where k is a positive integer. Find the value of k . 4. On April 1, 2001, there were 1280 members in a certain organization. On April 1, 2002,

the number of members in this organization had increased by 50%. On April 1, 2003, the number of members in this organization had increased by 50% from the number of members on April 1, 2002. If the membership continues to increase each year by 50%, how many members were in this organization on April 1, 2009?

5. If 3.5 pounds of chocolate cost $10.00, how many pounds of this same chocolate could be

purchased for $12.00 if the rate per pound is the same as that for the 3.5 pounds? Express your answer as an improper fraction reduced to lowest terms.

6. The quadratic equation 2 0x cx d+ + = has non-zero coefficients c and d . The roots for

x of this quadratic equation are also c and d . Find the value of ( )4 5c d+ . 7. Let k represent an even positive integer. One of the smallest eight values of k is

selected at random. Find the probability that the value selected is a positive integral multiple of 3. Express your answer as a common fraction reduced to lowest terms.

8. In a room there are only men and women. If 2 women were to leave, there would then be

twice as many men as women left in the room. If 14 men were then to leave, there would now be p times as many women as men left in the room. If p is a positive integer and there were a total of k people in the room at the start of this problem, find the sum of all distinct possibilities for the value of k .

WRITTEN AREA COMPETITION ALGEBRA I ICTM STATE 2010 DIVISION AA PAGE 2 OF 3

9. In the following number base problem, if 14

of 20X is 4, then what is 15

of 43X ?

10. Given the system: 3 8 2377817 14952

x yx ky+ =⎧

⎨ + =⎩. If k , x , and y are all positive integers, find the

value of k . 11. Let x and y represent two real numbers such that x y< . The difference between these

two numbers is 19, and the difference between the squares of these two numbers is 171. Find the smallest possible value of x such that the two numbers meet the given conditions.

12. Bonita Bondy invests k dollars in bonds at 6% interest per year and ( )80,000 k− dollars

in bonds at 5% interest per year. Renata Reverse invests ( )80,000 k− dollars in bonds at 6% interest per year and k dollars in bonds at 5% interest per year. Bonita’s total interest earned during the first year is $400 more than Renata’s total interest earned during the first year. Find the value of k .

13. The sum of the cubes of the roots for x of the equation 2 60 0x x k− + = is 84,420. Find

the larger of the two roots for x . 14. (Multiple Choice) For your answer write the capital letter which corresponds to the

correct choice.

If 0 2x< < , then 2 4 4x x− + is: A) 2x − B) 2 x− C) 2x + D) 2x− − E) x F) x−

Note: Be certain to write the correct capital letter as your answer.

WRITTEN AREA COMPETITION ALGEBRA I ICTM STATE 2010 DIVISION AA PAGE 3 OF 3

15. If x is a real number, find the maximum integral value of 2

25 25 925 25 38x xx x+ ++ +

.

16. The sum of two numbers that differ by 2 is 8p . In terms of p , the larger of the two

original numbers is kp w+ . If p , w , and k are all positive integers, find the smallest possible value of ( )3k w+ .

17. Six separate investments of $100 were made. One year later, each investment had

increased by an amount that was a member of the percentage set { }50%, 40%,30%, 20% . Let S be the dollar amount in the value of the sum of the six investments immediately after the increase. If each member of the percentage set was used at least once, find the sum of the values of all distinct possibilities for S .

18. (Multiple Choice) For your answer write the capital letter which corresponds to the

correct choice.

Which one of the following is never equal to the ratio of two whole numbers?

A) 23( )2

B) 0.13 C) 7% D) 3 E) 72

F) 3 272.1

Note: Be certain to write the correct capital letter as your answer. 19. An auto was originally priced at $10,000. The auto’s price was reduced by %x to y

dollars. Then the price of y dollars was reduced by %k to $5720. If both x and k are positive integers such that x k< , find the value of ( )2 3x k+ .

20. Let x and y be positive integers with y x> and with 71y < . Lee travels 100 miles at a

constant rate of x miles per hour and 100 miles at a constant rate of y miles per hour. Cindy also travels a total of 200 miles, but half of her time is spent at a constant rate of x miles per hour and the other half of her time is spent at a constant rate of y miles per hour. If Cindy’s average rate for the 200 miles is 4 mph more than Lee’s average rate for the 200 miles, find the sum of all distinct possibilities for y .

WRITTEN AREA COMPETITION GEOMETRY ICTM STATE 2010 DIVISION AA PAGE 1 OF 3 1. The ratio of the degree measures of the angles of a triangle is 5 : 6 : 7 . One of the angles

of this triangle is selected at random. Find the probability that the degree measure of the angle selected is an integral multiple of 3° . Express your answer as a common fraction reduced to lowest terms.

2. If the width of a rectangle is increased by 20% and the length of the rectangle is increased

by 25%, then the area of the rectangle is increased by %k . Find the value of k . 3. (Always, Sometimes, or Never) For your answer, write the whole word Always,

Sometimes, or Never—whichever is correct.

If the perimeter of a regular pentagon is equal to the perimeter of a square, then the sum of the lengths of the two diagonals of the square is greater than the perimeter of the regular pentagon.

4. Working at a constant rate, a girl finished 34

of a job in 12

of a day. Working at this

same constant rate, what fraction of a day will she require for the total job? Express your answer as a common fraction reduced to lowest terms.

5. (Multiple Choice) For your answer write the capital letter which corresponds to the

correct choice.

The equation of the line that contains the point ( )4,6− and is perpendicular to the x-axis is: A) 4x = − B) 4x = C) 6y = D) 6y = − E) 2x y+ =

Note: Be certain to write the correct capital letter as your answer. 6. In the diagram D lies on AB . ACD BCD∠ ≅ ∠ .

17AC = , 25BC = , and 28AB = . Find the area of ACDΔ .

7. The lengths of all sides of a triangle are integers. If two of the sides have respective

lengths of 13 and 32, find the probability that the length of the third side is a prime integer. Express your answer as a common fraction reduced to lowest terms.

8. The equation of the circle that passes through the points ( )2,4 and ( )1, 3− and that has its

center on the line 2 0x y+ = can be written in the form 2 2 2( ) ( )x h y k r− + − = where 0r > . Find the value of ( )h k r+ + .

C

BA D

WRITTEN AREA COMPETITION GEOMETRY ICTM STATE 2010 DIVISION AA PAGE 2 OF 3 9. In the diagram, there are 4 congruent circles that are each tangent

externally to two of the other circles as shown. The length of a diameter of one of the circles is 16. The area of the unshaded region enclosed by the circles can be expressed as k wπ− where k and w are positive integers. Find the smallest possible value of ( )k w+ .

10. Given the points ( )17, 21P − , ( )46,102Q , ( )14,6T − , and ( )2,19V . Let point R be

located between P and Q such that : 2 : 7PR RQ = . Find the area of RTVΔ . Express your answer as an exact decimal.

11. The perimeter of an isosceles triangle is 60. The length of the altitude to the base of this

isosceles triangle is 8. The length of an altitude to one of the legs of this isosceles

triangle can be expressed as kw

where k and w are positive integers. Find the smallest

possible value of ( )k w+ . 12. In the diagram, points A , B , C , and D lie on the

circle. C lies on AE , and D lies on BE . BC and AD intersect at F . If 50AFB∠ = ° and if

30CED∠ = ° , find the degree measure of minor arc CD .

13. Two identical right circular cones, each of height 3.103 are placed one on top of

another with each cone having its apex pointing downward and sharing the same vertical axis. That is, the bases of the cones are parallel and the apex of the top cone is at the center of the base of the bottom cone with the axis passing through both centers and both apexes. At the start, the top cone is full of water and the lower cone is empty. Water drips down, through a hole in the apex of the top cone, into the lower cone. Compute the height of water in the lower cone (height is measured from the apex of the lower cone to the center of the water level in the lower cone) at the moment when the height of the upper cone (measured from apex to the center of the water level) is 1.456. Assume the hole is small enough so that its radius can be disregarded for calculation purposes. Express your answer as a decimal rounded to the nearest thousandth.

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

Sometimes, or Never—whichever is correct.

If the diagonals of a parallelogram divide the parallelogram into four triangles of equal area, then the parallelogram is a square.

30°50°F

D

C

E

B

A

WRITTEN AREA COMPETITION GEOMETRY ICTM STATE 2010 DIVISION AA PAGE 3 OF 3 15. In the diagram, ABCD is inscribed in the circle. 12AB = ,

18BC = , 40CD = , and the area of ABCD is 9 4180759100

.

Find AD if 15AD < . Express your answer as a decimal. 16. A concave polygon has consecutive vertices on a graph of ( )2,7 , ( )7,7 , ( )7,10 ,

( )10,10 , ( )10,1 , and ( )2,1 . Find the area of this concave polygon. 17. In ADCΔ , 30ACD∠ = ° . B lies on AC such that

5AB = and 9BC = . Let E be a point on CD such that ( )AE EB+ is as small as possible. Find that value of

( )AE EB+ . Answer as a decimal rounded to the nearest hundredth.

18. How many of the following six choices are lines that have the equation 3x = ?

For your answer, write: 0, 1, 2, 3, 4, 5, or 6, whichever is correct. A) the line passing through ( )3, 4 and ( )3,32

B) the line passing through ( )3,0 , ( )3, 2 , and ( )3,6

C) the line passing through ( )3,3 that is parallel to the horizontal line 6y =

D) the line passing through ( )3,3 that is perpendicular to the horizontal line 7y =

E) the line passing through ( )3,3 that is perpendicular to the horizontal line 3y =

F) the line passing through ( )2,3− and ( )2,3 19. The graph of 6 8 0x y k− + = is tangent to the circle with a radius of 2 and a center at

( )12,11 . Find the value of k if 0k < . 20. Let k be a positive integer. A regular polygon of 6k sides is inscribed in a circle with a

radius whose length is 12. Let P be one of the vertices of the regular polygon. The sum of the squares of the distances from P to each of the other distinct vertices of the regular polygon is 31104. Find the area of the region that is in the interior of the circle but in the exterior of the regular polygon. Express your answer as a decimal rounded to the nearest thousandth.

C

A

D

B

A C

D

B

WRITTEN AREA COMPETITION ALGEBRA II ICTM STATE 2010 DIVISION AA PAGE 1 of 3 1. From the set { }1, 2,3, 4 , one member is selected at random as a value for k . Find the

probability that the solution for x of the equation 8kx = is an integer. Express your answer as a decimal.

2. (Multiple Choice) For your answer write the capital letter which corresponds to the

correct choice. What are all the real values for which 2x x x− = − − ? A) all 0x ≤ B) all 0x ≥ C) all 0x < D) all 0x > E) 0x = 3. Find the smallest possible positive value for x such that 22 9 5x x+ ≥ . 4. Find the integral value of x such that ( )( )( )9

2 3 5log log log 2x = .

5. If 3x − is a factor of 2 11x x k− + , find the value of k . 6. Find the focus of the parabola whose equation is 2 6 12 3 0x x y+ + − = . Express your

answer as an ordered pair of the form ( ),x y . 7. A machine contains 9 balls numbered consecutively from 1 through 9. Balls numbered 1,

4, 5, 6, 7, 8 and 9 are equally likely to drop out. Ball number 2 is four times as likely to drop out as any of these and three times as likely as ball number 3. The machine is rotated, and a ball drops out. The ball is replaced, the machine is rotated again, and a ball drops out. Find the probability that one 2 ball and one 3 ball dropped out. Express your answer as a decimal rounded to the nearest hundred-thousandth.

WRITTEN AREA COMPETITION ALGEBRA II ICTM STATE 2010 DIVISION AA PAGE 2 of 3 8. Let w be a positive integer that is a root for x of the cubic equation

3 2 1 02

x ax bx+ + − = . The reciprocal of w is also a root of the given cubic equation. If

50b < , find the largest possible value of w . 9. At the end of each month, Ernesto invests $125.02 in an annuity that pays an annual

percentage rate of 6.8% and is compounded monthly. After Ernesto has made his 24th monthly investment of $125.02, find the number of dollars in the value of Ernesto’s annuity. Round your answer to the nearest dollar, and express your answer as that whole number.

10. Let 57n < and let x n< where both x and n are positive integers. Find the value of the largest possible value of x if the sum of the consecutive positive integers from ( )1x + through n is 186 more than the sum of the consecutive positive integers from 1

through ( )1x − . 11. The foci of a hyperbola are ( )0,3 and ( )0, 3− , and the vertices of this hyperbola are

11(0, )2

and 11(0, )2

− . This hyperbola can be expressed in the form 2 2

1y xk w− = . Find

the value of ( )5 9k w+ . 12. Find the sum of the arithmetic series 2 5 8 11 302+ + + + + . 13. A six digit number of the form bcdefa , where b is the hundred thousands digit, c is the

ten thousands digit, etc., is 3.5 times the six digit number of the form abcdef , where a is the hundred thousands digit, etc. Find the original six digit number bcdefa .

14. Let 1 125a = . For every integer n such that 1n ≥ , ( )1 nna a+ = + the square of the sum of

the digits of na . Find the value of 6a .

WRITTEN AREA COMPETITION ALGEBRA II ICTM STATE 2010 DIVISION AA PAGE 3 of 3 15. If 3 39 12 9xy x y− = + , then the range for y includes all real numbers except one real

number. Find the real number that is not a member of the range for y . 16. On 11/14/01, Michael Cameron, 20, from Canada proved that 13,466,9172 1− is prime. How

many digits did that prime number contain? 17. An ordered pair ( ),b c of integers, each of which has absolute value less than or equal to

nine, is chosen at random, with each such ordered pair having an equal likelihood of being chosen. The individual members of the ordered pair chosen are substituted for b and c respectively in the equation 2 2 0x bx c+ + = . Find the probability that the two roots for x are unequal. Express your answer as a common fraction reduced to lowest terms.

18. If 5x ≠ , then 3

22 20 5 235 5

x x kx xx x− +

= + + +− −

. Find the value of k .

19. If 1x > and if ( )( )128log 128 log 16x y= , then ( )2log 256xky

wy p=

+ where k , w , and p

are positive integers. Find the smallest possible value of ( )k w p+ + .

20. Let x , y , and z be positive integers such that 2 2 21 1 1x y z

+ = . If x and y are each

integral multiples of P , a prime integer greater than 6, find the smallest possible value of ( )x y z+ + .

WRITTEN AREA COMPETITION PRECALCULUS ICTM STATE 2010 DIVISION AA PAGE 1 OF 3 1. Let k be selected at random from the set { }1, 2,3, 4,6,8,12,18 . If 3x k= , find the

probability that the solution for x is a whole number. Express your answer as a common fraction reduced to lowest terms.

2. Let x be a positive integer such that 100x < . For how many distinct values of x is

sin(2 ) 1 2x + > ? 3. If the sides of a triangle have lengths of 3a , 2b , and c , with A opposite side of length

3a , then 2 2 2 cos( )pa qb rc cwb A= + − . If p , q , r , and w are all positive integers, find the smallest possible value of ( )p q r w+ + + .

4. If ( ) 3 4f x x= − , then an equation for the inverse can be written as 1( ) x kf xw

− += . Find

the value of ( )k w− . 5. If x is a positive integer, find the smallest possible value of x such that cot( )x° and

cot( )x− ° are equal numbers. 6. From physics, the height s , in feet, of an object after t seconds that has been thrown

straight up from a point os feet above ground level with an initial velocity of ov is 216 o os t v t s= − + + . If a baseball is thrown straight up from 3 feet above ground level

with initial velocity of 97 feet per second, find the number of feet in the maximum height that the ball will reach. Express your answer as a decimal rounded to the nearest hundredth of a foot.

7. A right circular cylinder has a volume of 12π . The lengths of the radius ( )r of the base

and the height ( )h of the cylinder are both positive integers. Find the sum of all possible

distinct values of ( )r h+ where r and h represent the radius and height of one cylinder. 8. The probability that Mary will have one of the top ten Precalculus scores is 0.4263. The

probability that Peter will have one of the top ten Precalculus scores is 0.3642. The probability that both will have scores among the top ten Precalculus scores is 0.1968. Expressed as a decimal rounded to 4 significant digits, find the probability that both had scores in the top ten Precalculus scores if it is known that at least one of the two did.

9. Rounded to the nearest year, find the number of years needed for a sum of money to

quadruple if invested at 7% annual percentage rate compounded continuously. 10. In ABCΔ , 7AB = , and 8AC = . ( )cos BAC∠ is a rational number that can be expressed

as kw

where k and w are relatively prime positive integers. It is known that ( )k w+ is

the square of a positive integer and that BC is a positive integer. Find the sum of all possible distinct values of BC .

WRITTEN AREA COMPETITION PRECALCULUS ICTM STATE 2010 DIVISION AA PAGE 2 OF 3 11. A box that contains exactly 35 yellow marbles, 7 green marbles, and 8 red marbles.

From this box, 5 marbles are drawn (without replacement) at random. Find the probability that the 5 marbles drawn were exactly 3 yellow, 1 green, and 1 red. Express your answer as a common fraction reduced to lowest terms.

12. (Multiple Choice) For your answer write the capital letter which corresponds to the

correct choice. Given: A parabola whose points are all equidistant from the point ( )2,5

and the line whose equation is 7x = and that has a latus rectum of length 16 5 .

A) The conditions above determine a conic. B) The conditions above overdetermine a conic (that is, there is no conic meeting

all the given conditions). C) The conditions above underdetermine a conic (that is, there is more than one

conic that meets all the given conditions). Note: Be certain to write the correct capital letter as your answer.

13. On the circle whose equation is 2 2( 25) ( 30) 145x y− + − = a particle is at ( )26, 42A and

another particle is at ( )24,18B . At the same instant, both particles begin to move on the circle. The particle at A moves at a constant speed counter-clockwise on the circle, and the particle at B moves at a constant speed clockwise on the circle. The two particles meet for the first time at ( )17,39 . The particle that started at A is moving k times the rate of the particle that started at B . Find the value of k . Express your answer as a decimal rounded to the nearest hundredth.

14. In this problem, assume that the standard deviation is calculated according to the

standard method of calculating the standard deviation for a set of sample proportions. Also, assume the following table of z-scores with the accompanying standard normal probabilities is accurate. On the table – 2(.0228) means that there is a normal probability of .0228 of obtaining a z-score of 2− or less.

– 2(.0228) – 1.9(.0287) – 1.8(.0359) – 1.7(.0446) – 1.6(.0548) – 1.5(.0668) – 1.45(.0735) – 1.4(.0808) –1.35(.0885) – 1.3(.0968) – 1.25(.1056) – 1.2(.1151) – 1.1(.1357) –1(.1587) – 0.9(.1841) – 0.8(.2119) – 0.75(.2266) – 0.7(.2420) – 0.6(.2743) – 0.5(.3085) – 0.4(.3446) – 0.3(.3821) – 0.25(.4013) – 0.1(.4602) 0(.5000) 0.1(.5398) 0.2(.5793) 0.25(.5987) 0.3(.6179) 0.4(.6554) 0.5(.6915) 0.6(.7257) 0.7(.7580) 0.75(.7734) 0.8(.7881) 0.9(.8159) 1.0(.8413) 1.1(.8643) 1.2(.8849) 1.25(.8944) 1.3(.9032) 1.4(.9192) 1.5(.9332) 1.6(.9452) 1.7(.9554) 1.75(.9599) 1.8(.9641) 1.9(.9713) 2.0(.9772) 2.1(.9821) 2.2(.9861) 2.25(.9878) 2.3(.9893) 2.4(.9918) 2.5(.9938) Assume that 80% of all mathletes know the value of 3 to 4 significant digits. In a sample observation, Carol selects 100 mathletes at random. Find the probability that the proportion of those mathletes who know the value of 3 to 4 significant digits is between 76% and 82%. Express your answer as a decimal rounded to 4 decimal places.

WRITTEN AREA COMPETITION PRECALCULUS ICTM STATE 2010 DIVISION AA PAGE 3 OF 3 15. Let a , b , and c be positive integers. Find all ordered triples ( ), ,a b c such that

3 4 7 1482 1 8 157

abc

⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥ =⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦⎢ ⎥⎣ ⎦

. Be certain to express your answer as ordered triples of the

form ( ), ,a b c . 16. If 4 215 36x x− + is factored over the reals into four linear factors, one of those linear

factors, in simplest form, is x k w− where k and w are positive integers greater than one. Find the value of ( )5 3k w+ .

17. If a stick is broken at random into 3 pieces, find the probability that the longest of the 3

pieces is at least three times the shortest of the 3 pieces and that the 3 pieces can serve as the 3 sides of a triangle. Express your answer as a common fraction reduced to lowest terms.

18. If 2

2 0( )

1 0x if x

f xx if x− ≤⎧

= ⎨ + >⎩, then the range of f can be expressed in interval notation as

( , ] ( , )k w−∞ ∞∪ . Find the value of ( )2 11k w+ . 19. The coordinate axes of the graph of the equation 2 27 8 6 5 28 0x xy y+ − − = are rotated

through a positive acute angle ( )θ so as to eliminate the xy term. Find the value of sin( )θ . Express your answer as a decimal rounded to the nearest ten-thousandth.

20. In the figure shown, 15AB AC= = , 3BD = , and

19DC = . Points A , E , and B are collinear; points B , D , and C are collinear; and points A , F , and C are collinear. AE DE= , and AF DF= . Find DE . Express your answer as a decimal rounded to the nearest hundredth.

3 19B C

A

D

EF

FROSH-SOPH EIGHT PERSON TEAM COMPETITION ICTM STATE 2010 DIVISION AA PAGE 1 of 3

NO CALCULATORS

NO CALCULATORS

1. Let { }4,6, 2,1,5,18C = − . If one of the members of C is selected at random and

substituted for x , find the probability that 2 3 14x + < . Express your answer as a common fraction reduced to lowest terms.

2. Find the number of feet in the height of a tree if the tree is 96 feet shorter than the height

of a building whose height is 5 times that of the tree. 3. The lengths of the sides of a scalene triangle are in the ratio of 3 : 4 : k . If k is an

integer, find the largest possible value of k . 4. A triangle has two sides of respective lengths 4 and 5. The length of the third side of the

triangle is an integer. If a triangle is selected at random from one of the triangles meeting the given conditions, find the probability that the triangle is obtuse. Express your answer as a common fraction reduced to lowest terms.

5. Two boys, A and B, start at the same time to ride on the same road from Champaign to

Peoria, a distance of 60 miles. A travels at a constant rate that is 4 miles per hour slower than B’s constant rate. B reaches Peoria and immediately turns back meeting A 12 miles from Peoria. Find the number of miles per hour in A’s constant rate.

6. The ratio of the degree measures of the angles of a triangle is 1: 2 : 3 . The lengths of two

of the sides of the triangle are 4 and 8. A solid is generated by rotating this triangle one

revolution about the third side. The volume of this solid can be expressed as k wp

π

where k , w , and p are positive integers. Find the smallest possible value of ( )k w p+ + .

7. A right circular cylinder has the same volume as a right circular cone. Both figures also

have the same lateral area. Let h represent the height of the cylinder, k represent the height of the cone, r represent the radius of the circular base of the cylinder, and x

represent the radius of the circular base of the cone. Then 2 2

22 29

ar kxbk r

=−

where a and

b are positive integers. Find the value of ( )a b+ . 8. A circle has an equation of 2 2 6 8 16 0x y x y+ − + − = . The circumference of this circle

can be expressed in the form k wπ where k and w are positive integers. Find the smallest possible value of ( )k w+ .

FROSH-SOPH EIGHT PERSON TEAM COMPETITION ICTM STATE 2010 DIVISION AA PAGE 2 of 3

NO CALCULATORS

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P

R

S

Q

T

V

9. If 8 is 20% of x , find the value of x . 10. Let 5x y+ = and 3 3x y k+ = . If both x and y are real numbers, find the smallest

possible value of k . Express your answer as a decimal. 11. In the diagram, points P, Q, and R are collinear, points P, T, and S are

collinear, and points Q, V, and T are collinear. PR and PS are tangent to the circle at R and S respectively, and QT is tangent to the circle at V. (Always, Sometimes, or Never) For your answer, write the whole word Always, Sometimes, or Never—whichever is correct for the following statement: If 18PR = , then the perimeter of PQTΔ is 36.

12. The graphs of 2 27 84 105 28 7 0x y x y− + + + = and 1y x= − intersect in two points.

The y-coordinate of one of those points is 2

k w+ where k and w are positive integers.

Find the value of ( )k w+ . 13. (Multiple Choice) For your answer write the capital letter which corresponds to the

correct choice.

If x is a real number, then the set of values for x such that 7 7x x− < − is: A) ∅ B) { }: 7x x ≠ C) { }: 7x x ≠ − D) { }: 0x x ≤ E) { }: 0x x ≥

F) { }: 0x x < G) { }: 0x x > H) { }: 7x x < J) { }: 7x x > − Note: Be sure to write the correct capital letter for your answer.

14. A regular icosahedron is a solid figure, a polyhedron whose twenty faces are congruent

equilateral triangles, with each face of a standard icosahedral die being equally likely to come up when the die is rolled. Let the faces of a standard icosahedral die be numbered with the 20 integers from 1 to 20 inclusive. If two fair, standard icosahedral dice are thrown, find the probability that the sum of the numbers on the uppermost faces is 21. Express your answer as a common fraction reduced to lowest terms.

FROSH-SOPH EIGHT PERSON TEAM COMPETITION ICTM STATE 2010 DIVISION AA PAGE 3 of 3

NO CALCULATORS

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15. A , B , C , and D are four people who live in a land where each person always makes

true statements or always makes false statements. They state: A : B is a liar. B : Both C and D are liars. C : At least three of us four are liars. D : At least one of A and B is a liar. Using T for truthteller and L for liar, write the ordered quadruple that describes ( ), , ,A B C D . For example, if A , B , and C are truthtellers and D is a liar, your ordered

quadruple would be ( ), , ,T T T L . 16. Let k and w be two positive integers with k w> and 30w ≠ . If 2 2 2 230 40 k w+ = + ,

find the value of ( )2 5k w+ . 17. One 12 hour clock is running at a constant gain of 6 minutes per 24 hours. A second 12

hour clock is running at a constant loss of 3 minutes per 8 hours. The first clock now shows a time of 12:00, and the second clock now shows a time of 1:30. What time will be showing on these two clocks when these two clocks show identical times the second time from now? Express your answer in the form hours:minutes . Note: “identical times” means, for example, if one clock shows 9:20, then the other clock will also show 9:20.

18. The area of a right triangle is 24 square units. If one of the two shorter sides of the right

triangle are has a length of 6 units, find the number of units in the length of the hypotenuse.

19. In the diagram ABCD is a square with

X , A , B , and Y being collinear points. The square rolls, without sliding, along XY turning about B , C , and D in succession. If 3AB = , the length of the

path of point A until it again lies on XY can be expressed as k w f

pπ π+

where k , w ,

f , and p are positive integers. Find the smallest possible value of ( )k w p f+ + + . 20. How many distinct four digit numbers are integral multiples of 11 if no digit is equal to

zero and no two digits of any four digit number are alike?

D

A

C

BX Y

JUNIOR-SENIOR EIGHT PERSON TEAM COMPETITION ICTM STATE 2010 DIVISION AA PAGE 1 OF 3

NO CALCULATORS

NO CALCULATORS

1. Let 1 2, , 2,174 3

A ⎧ ⎫= −⎨ ⎬⎩ ⎭

. If one of the members of A is selected at random and

substituted for x , find the probability that 2x x> . Express your answer as a common fraction reduced to lowest terms.

2. One of the angles of a parallelogram has a degree measure of 50. If two of the angles of

the parallelogram are selected at random without replacement, find the probability the two angles selected are congruent. Express your answer as a common fraction reduced to lowest terms.

3. Find the smallest real value of x such that ( )2log 8 8 0x x+ − = . 4. Each of the three circles shown in the diagram is tangent to the other two.

If the radii of the three circles have lengths of 3, 5, and 11, find the perimeter of the triangle formed by connecting the centers of the three circles.

5. Bob and Judy are playing a game with four dice whose faces are numbered as follows:

a. Orange Die: 4, 4, 4, 4, 4, 4 b. Blue Die: 8, 8, 2, 2, 2, 2 c. Scarlet Die: 7, 7, 7, 1, 1, 1 d. Gray Die: 6, 6, 6, 6, 0, 0 Judy must select a die first. After Judy selects her die, but before she rolls, Bob must select his die. Each then rolls once. The winner is the one who rolled the higher number. If Bob applies best strategy, find the probability that Bob will win the game. Express your answer as a common fraction reduced to lowest terms.

6. Let 1i = − . The roots for x of a quadratic equation are 7 3i− and 7 3i+ . This

quadratic equation can be expressed in the form 2 0x kx w+ + = where k and w are integers. Find the value of ( )2k w+ .

7. (Multiple Choice) For your answer write the capital letter that corresponds to the

best answer. Which one of the following statements is true? A) ( )x A x A B∈ → ∈ ∩ B) ( ')x A x A B∈ → ∈ ∩ C) x A x B∈ → ∈ D) ( )( )x A x B A B A∈ → ∈ ∩ ∪ ∪ E) ( )( ) 'x A x B A B A∈ → ∈ ∩ ∪ ∪

Note: Be sure to write the correct capital letter as your answer.

JUNIOR-SENIOR EIGHT PERSON TEAM COMPETITION ICTM STATE 2010 DIVISION AA PAGE 2 OF 3

NO CALCULATORS

NO CALCULATORS

8. Alice, Betsy, Carl, Dick, Ethel, and Frank were the only people entered in a round robin

tournament in which each person plays each other person in a “3 game match.” In a “3 game match”, the match consists of 2 games if the same person wins the first two games, and the match consists of 3 games if the two people playing each win one of the first two games. Find the maximum number of games that can be played in this round robin tournament.

9. How many distinct non-real roots does the equation 3 198 6754 0x x+ + = have? 10. Let {111,114,118,123,129,136,144,153,163,174}x∈ . Find the sum of all distinct values

of x such that 4 3 26 11 10

12x x x x+ + + is an integer.

11. Find 3

2

8lim( )2x

xx→

−−

.

12. Let ( 1)( 1)

1 3 5 7 9 2 1 ( 1)3 6 12 24 48 3(2 )

nn

nx +−

−= − + − + + ⋅ ⋅ ⋅ + − + ⋅ ⋅ ⋅ . Find the value of x .

Express your answer as a common fraction reduced to lowest terms. 13. On n successive tests, Tom received the following scores: 1 3 5 7 (2 1), , , , , nx x x x x −⋅⋅⋅ . The

arithmetic mean of Tom’s scores can be expressed in sigma notation as ( )1

hn

jk mk

f xn +

=∑ .

Find the value of ( )f h j m+ + + . Assume that f , h , j , and m are constants. 14. On a table are 3 red marbles, 4 gray marbles, and 2 yellow marbles. Martha removes one

of the 9 marbles at random from the table. If Martha’s selection was yellow, Mark selects a marble at random from the red and gray marbles. If Martha’s selection was gray, then Mark selects a marble at random from the red and yellow marbles. If Martha’s selection was red, Mark then selects at random one of the 8 remaining marbles. Find the probability that Mark selection was a red marble. Express your answer as a common fraction reduced to lowest terms.

JUNIOR-SENIOR EIGHT PERSON TEAM COMPETITION ICTM STATE 2010 DIVISION AA PAGE 3 OF 3

NO CALCULATORS

NO CALCULATORS

15. The eccentricity of an ellipse whose equation is 2 2

13 75x y

+ = can be expressed in the form

k wp

where k , w , and p are positive integers. Find the smallest possible value of

( )k w p+ + .

16. Find the integer that is equivalent to 7( )3

1( )4

log 32(8 )−⎛ ⎞

⎜ ⎟⎝ ⎠

.

17. In a room with exactly 9 distinct persons, exactly 30 handshakes between 2 distinct

persons were made. No two of these handshakes took place between the same 2 persons. Of the persons in the room, three, and only three, persons shook hands with six, and only six, other persons in the room. Of the remaining persons in the room, two, and only two, persons shook hands with seven, and only seven, other persons in the room. Those two persons each failed to shake the hands of the same person in the room. Three distinct persons from these 9 persons are selected at random. Find the probability that those three distinct persons had all shaken hands with each other from among the 30 handshakes. Express your answer as a common fraction reduced to lowest terms.

18. Let 1i = − . The complex conjugate of 4 18i − is k i w+ . Find the value of ( )2 3k w+ . 19. In the order shown below, points A, B, C, D, E, and F are located on a number line. The

coordinate of point A is zero. The coordinates of B, C, D, E, and F are all positive integers. The distance between any two of the six points is a unique positive integer (in other words, no two distances are the same). Find the smallest possible coordinate of point F. A B C D E F

0 20. Let a , b , and c be positive integers such that a b c< < . Let a , b , and c be the lengths

of the sides of Triangle ABC (with a opposite A∠ , etc.) in which cos( ) cos( ) cos( ) 151

990A B C

a b c+ + = . If 2a , 2b , and 2c are the roots for x of the cubic

equation 3 2302 0x x kx w− + − = , find the smallest possible value of c .

CALCULATING TEAM COMPETITION ICTM STATE 2010 DIVISION AA PAGE 1 of 3 Answers should be expressed either in scientific notation OR as a decimal, and answers should be rounded to four significant digits. However, specific instructions in a given problem take precedence. For example, if instructions ask for the answer to be expressed as a decimal or as an integer, you may NOT use scientific notation for that answer.

1. In the year of 2008, Pop drank an average of 4261

cans of soda each day. Find the total

number of cans of soda that Pop drank during the year of 2008. Express your answer as an integer.

2. On a 10 problem test, Lee answered three of the first five at random, leaving two blank,

and two of the last five at random, leaving three blank. Cindy answered two of the first five at random, leaving three blank, and three of the last five at random, leaving two blank. By how much does the probability that Lee answered question #8 exceed the probability that Cindy answered both questions #1 and #10? Express your answer as a common fraction reduced to lowest terms.

3. The perimeter of an isosceles triangle ABC is 585.6. If BC is ( )0.706 AC , find the

largest possible value of BC . 4. From a point on horizontal level ground at the end of the shadow of a vertical flagpole,

the angle of elevation to the top of the flagpole is 26.13° . If the shadow is 98.62 feet, find the number of feet in the height of the flagpole.

5. Let k be the length of the altitude of integral length of a triangle whose sides have

lengths of 11, 25, and 30. Let w be the length of the altitude of integral length of a triangle whose sides have lengths of 22, 85, and 91. Find the value of ( )k w+ . Express your answer as an integer.

6. A rectangle whose length is 56.47 and whose width is 34.56 is inscribed in a circle. Find

the area of the circle. 7. If 4.789 is a radian measure, find sec(4.789) . 8. ABCΔ has vertices at ( )8.742,8.742A , ( )12.68,14.48B , and ( )28.22,14.48C . The

vertical line whose equation is of the form x k= divides the triangle into two regions of equal area. Find the value of k .

9. How many non-congruent triangles exist such that two sides have lengths of 62.48 and

89.72 and such that the angle opposite the side of length 62.48 is 33 16 '° ? Express your answer as an integer.

CALCULATING TEAM COMPETITION ICTM STATE 2010 DIVISION AA PAGE 2 of 3 10. If 90 180x° < < ° , find, to the nearest tenth of a degree, the value of x such that

23sin ( ) 4.003 5.149cos( )x x= + . 11. Head is the pressure consumed (that is, lost) in forcing a fluid along against the

resistances caused by the conduits through which the fluid flows. The equation for the head lost by friction in a water pipe is:

20.01739 0.1087(0.0126 )

2d LVH

gdV−

= + where

H = number of meters in the loss of head due to friction V = number of meters per second of the velocity inside the pipe L = number of meters in the length of the pipe g = gravitational constant of 9.800 meters per second squared d = number of meters in the inner diameter of the pipe If 0.4987V = , 5.001L = , and 0.01122d = , find the number of meters in the loss of head due to friction.

12. Central angle CAB of a circle is 63.37° and the area of the circle is 84.26π . If A is the

center of the circle, and C and B lie on the circle, find CB . 13. In this problem, assume that the standard deviation is calculated according to the standard

method of calculating the standard deviation for a set of sample proportions. Also, assume the following table of z-scores with the accompanying standard normal probabilities is accurate. On the table 0.1(.53988) means that there is a normal probability of .5398 of obtaining a z-score of 0.1 or less.

– 0.8(.2119) – 0.75(.2266) – 0.7(.2420) – 0.6(.2743) – 0.5(.3085) – 0.4(.3446) – 0.3(.3821) – 0.25(.4013) – 0.1(.4602) 0(.5000) 0.1(.5398) 0.2(.5793) 0.25(.5987) 0.3(.6179) 0.4(.6554) 0.5(.6915) 0.6(.7257) 0.7(.7580) 0.75(.7734) 0.8(.7881) 0.9(.8159) 1.0(.8413) 1.1(.8643) 1.2(.8849) 1.25(.8944) 1.3(.9032) 1.4(.9192) 1.5(.9332) 1.6(.9452) 1.7(.9554) 1.75(.9599) 1.8(.9641) 1.9(.9713) 2.0(.9772) 2.1(.9821) A sample survey in Urbana revealed that Urbana citizens eat an average of 700 chocolate bars per year with a standard deviation of 110 chocolate bars and that the results were normally distributed. Assume that a new survey of 1000 Urbana citizens was taken and that the results were normally distributed. Rounded to the nearest whole number, find the number of Urbana citizens in this new survey that could be expected to eat between 645 and 810 chocolate bars per year. Express your answer as an integer.

CALCULATING TEAM COMPETITION ICTM STATE 2010 DIVISION AA PAGE 3 of 3 14. Carbon-14, an isotope of carbon, has a half-life of 5750 years. A mummy discovered in

the pyramid Hegippedyou had lost 48.11% of its carbon-14. Rounded to the nearest year, find the age of the mummy. Express your answer as an integer.

15. Each of the two lower base angles of an isosceles trapezoid has a measure of 76.38° .

The lower base has a length is 7.986, and each leg each has a length of 6.012. Find the length of a diagonal of the isosceles trapezoid.

16. The diagonal distance of a rectangular television screen is 36.00 inches. The radian

measure of the angle formed by the diagonal and the longer side of the screen is 0.6643. Find the number of inches in the length of the longer side of the television screen.

17. A baseball field has its bases 90 feet apart in the form of a square. Speedy is on first base

and has a “lead” of 9.000 feet toward second base. Before Fireball, the pitcher, releases the pitch, Speedy gets an additional “jump” of 6.000 feet toward second base as he runs toward second base. Fireball’s pitch is timed at 92.48 mph. and travels 59.00 feet after its release until it is caught by Rifle, the catcher. Rifle needs 1.000 seconds before he releases his throw from exactly over the middle of home plate to second base. Rifle’s throw to second base is timed at 82.15 mph. Speedy’s running rate is 94.08 feet per 3.597 seconds. If the second baseman catches the ball directly over the middle of second base, find the number of seconds the second baseman will have the ball before Speedy arrives at the middle of second base. Express your answer as a decimal rounded to the nearest hundredth of a second.

18. Regular polygon ABCDEFGHIJKLMNOPQRSTUVW is inscribed in a circle. Find the

degree measure of arc ABC . 19. The length of a diagonal of a square is 200 . In the same plane, a circle is drawn with

the center of the circle at the intersection of the diagonals of the square. The length of the radius of the circle is either 1, 2, 3, 4, or 5 with each of these lengths being equally likely. If a point is picked at random in the interior of the square, find the probability that the point that is picked is in the exterior of the circle. Express your answer as a decimal rounded to the nearest thousandth.

20. On a plane, a bug starts at ( )15, 20A . The bug must travel to the line represented by

5 80y x= − + to ( )4,7B , and to ( )2,1C . However, the bug has its choice as to which of the three objects it will travel to first, which object will be second, and which object will be third. Find the next to largest possible total distance for any particular order that the bug can travel if it starts at A and travels, in some order, to each of the other three objects. Assume that whenever the bug chooses any particular order of travel to the three objects that the bug always travels the shortest possible total distance for that particular order.

FROSH-SOPH 2 PERSON COMPETITION ICTM 2010 STATE DIVISION AA PAGE 1 OF 1 1. By how much does the length of a diagonal of a square whose perimeter is 144 exceed

the length of a diagonal of a rectangle whose two dimensions are of respective lengths of 3 3 and 5 ?

2. For all real numbers a and b , ( 5)( 3)a b a b⊕ = − + . Let p be the perimeter of a square

whose diagonal has a length of 648 . Find the value of 9 11 p⊕ + . 3. If 7x + is equal to the units digit of 3013 , find the value of x . 4. Let r be the length of the radius of the inscribed circle of a triangle whose side-lengths

are 10, 24, and 26. Let w be the slope of a line that is perpendicular to the line passing through ( )15.2,16.8 and ( )0.8, 20.4 . Find the value of ( )r w+ .

5. ABCD is a quadrilateral in which the diagonals are perpendicular bisectors of each other.

It is known that ABC∠ has a degree measure of 120, and it is known that one of the diagonals of quadrilateral ABCD has a length of 24. Find the largest possibility for the perimeter of quadrilateral ABCD .

6. Let w and k be single digit positive integer digits in the following number base problem

110eight eleven tenwkw kwk w+ = . Find the value of ( )2 3k w+ . Express your answer in base ten.

7. The line through point ( )2.10,10.24P − and parallel to the line 4.72 1.55 2010x y− + =

has a y-intercept whose graph has coordinates ( )0,Q b . Find the distance PQ . Express your answer as a decimal rounded to the nearest hundredth.

8. Let k be the value of x when ( )( )( )2 3 21 2 1 3 3 1 123x x x x x x+ + + + + + = . Let w be the

value of x when ( )( )( )2 3 21 2 1 3 3 1 123x x x x x x− − + − + − = . Find the largest possible

value of ( )k w+ . Express your answer as a decimal rounded to the nearest hundredth. 9. Let k be the number of distinct terms in the simplified expansion of ( )3a b c+ + . Find

the sum of the numeric coefficients in the expansion of ( )kx y+ . 10. In a right triangle with the altitude drawn to the hypotenuse, the following five lengths

are all integers: both legs, the altitude to the hypotenuse, and both portions of the hypotenuse into which the altitude splits the hypotenuse. If one of these five lengths is 90, find the smallest possible sum of the other four lengths.

2010 SAA School ANSWERS

Fr/So 2 Person Team (Use full school name – no abbreviations) Total Score (see below*) =

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.

Answer Score (to be filled in by proctor) 1. 2. 3. 4. 5. 6. (Must be this decimal.) 7. (Must be this decimal.) 8. 9. 10. TOTAL SCORE: (*enter in box above) Extra Questions: 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 first In round with correct answer

32 2

128 4− 8

96 28 OR 1028 OR 28ten

6.73 4.46

1024 342

NOTE: Questions 1-5 only are NO CALCULATOR

JUNIOR-SENIOR 2 PERSON COMPETITION ICTM 2010 STATE DIVISION AA PAGE 1 OF 1

1. Find the units digit of 789 78913 7+ . 2. Let k be an integer. There is a root for x that is larger than 2− but less than 1− for the

equation 43 11 0x kx+ − = . Find the largest possible value of k . 3. The number H , when expressed in base seven, is 0. 2seven where the 2 repeats. When

expressed in base ten, kHw

= where k and w are relatively prime integers. Let p be

the sixteenth term of the arithmetic progression whose 2nd term is 12 and whose fourth term is 19. Find the value of ( )k w p+ + .

4. The area of an ellipse whose equation is 2 2

112x y

k+ = is 24π . Find the value of k .

5. A set of consecutive positive integers has at least two members and the sum of those

members of the set is 36. How many such distinct sets are there? 6. Let c , d , and e be positive integers such that no two of them are consecutive integers.

Let ( ),C n r be a symbol for combinations of n distinct things taken r at a time. If

( ) ( )1, 2 , 2 13C c C c+ − = and if ( ) ( ), 2 , 2 13C e C d− = , find the value of ( )3 2c d e+ + . 7. Let { }38,97, 42,78,63,12,19, 25,96,93,87,81,56A = . Let k be the number of distinct

members of A for which the z-score is negative. Let 2x , 43, and 47x + taken in that order form an arithmetic sequence. Find the value of ( )k x+ .

8. Let 120!k = Written in base ten, k ends in 28 zeroes; written in base seven, k ends in

19 zeroes. In how many zeroes will k end if k is written in base twenty-four? 9. Let k be the number of distinct positive integral factors of the greatest common factor of

4104 and 7632. If w is a positive integer, let S be the sum of all distinct values of w such that 2log 6w < . Find the value of ( )k S+ .

10. Let x be the largest integer which leaves the same remainder when dividing each of the

three numbers 163, 305, and 518. Let y be the fourth term in the sequence defined recursively as follows: the first term 1 2a = and ( 1) 5 3n na a+ = − if 1n ≥ . Find the

value of ( )x y+ .

2010 SAA School ANSWERS

Jr/Sr 2 Person Team (Use full school name – no abbreviations) Total Score (see below*) =

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.

Answer Score (to be filled in by proctor) 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. TOTAL SCORE: (*enter in box above) Extra Questions: 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 first In round with correct answer

0 18 65 48 2

59 19 38

2028 228

NOTE: Questions 1-5 only are NO CALCULATOR

FRESHMAN-SOPHOMORE RELAY COMPETITION ROUND 1 ICTM 2010 DIVISION AA STATE FINALS QUESTION 1 1. Rounded to the nearest tenth, find the smallest possible value of k such that

( ) ( )7.47 3 2.46 2.78 0.26k k− − + > . Express your answer as a decimal.

FRESHMAN-SOPHOMORE RELAY COMPETITION ROUND 1 ICTM 2010 DIVISION AA STATE FINALS QUESTION 2 2. Working at a constant speed, Glen can mow a certain lawn in 57.3 minutes.

Working at a constant speed, Brooke can mow the same lawn in ANS fewer minutes than Glen. After Glen has been mowing this lawn for 21.0 minutes, he is relieved by Brooke who finishes the job alone. Find the number of minutes Brooke must work to finish mowing this lawn. Express your answer as a decimal rounded to the nearest tenth.

FRESHMAN-SOPHOMORE RELAY COMPETITION ROUND 1 ICTM 2010 DIVISION AA STATE FINALS QUESTION 3 3. Two angles of a convex pentagon are angles with respective degree measures of

80.1° and 132.6° . The respective degree measures of the third, fourth, and fifth angles of this pentagon are x° , y° , and z° . If y x ANS= + and z y ANS= + , find the degree measure of the smallest angle of this pentagon. Express your answer as a decimal.

FRESHMAN-SOPHOMORE RELAY COMPETITION ROUND 1 ICTM 2010 DIVISION AA STATE FINALS QUESTION 4 4. A right circular cylinder has a base whose diameter is ANS and whose height is

16.1. Find the volume of this right circular cylinder. Express your answer as a decimal rounded to the nearest tenth.

FRESHMAN-SOPHOMORE RELAY COMPETITION ROUND 2 ICTM 2010 DIVISION AA STATE FINALS QUESTION 1 1. Let ( )x kwpf f g= + . If k , w , p , and f are distinct positive prime odd

integers, and g is an integer greater than ( )k w p f+ + + , find the smallest possible value of x .

FRESHMAN-SOPHOMORE RELAY COMPETITION ROUND 2 ICTM 2010 DIVISION AA STATE FINALS QUESTION 2 2. How many of the following four equations have a solution that is an integral

factor of ANS ? For your answer, write 0, 1, 2, 3, or 4, whichever is correct.

A) 15( 7) 354( 6) 9

xx+

=+

B) ( )7.846 3 235.38x − = C) ( )8 1 5x− − =

D) ( )16 3 2 7x− + = −

FRESHMAN-SOPHOMORE RELAY COMPETITION ROUND 2 ICTM 2010 DIVISION AA STATE FINALS QUESTION 3 3. The length of the median drawn to the hypotenuse of a 30° , 60° , 90° triangle is

ANS . Find the length of the altitude to the hypotenuse of this triangle. Express your answer as a decimal rounded to the nearest thousandth.

FRESHMAN-SOPHOMORE RELAY COMPETITION ROUND 2 ICTM 2010 DIVISION AA STATE FINALS QUESTION 4 4. In a circle whose radius has a length of 8.312 units, two parallel chords are drawn

that are ANS units apart. The longer chord is 4.214 units from the center of the circle. Find the number of units in the length of the shorter of the two chords. Express your answer as a decimal rounded to the nearest thousandth.

FRESHMAN-SOPHOMORE RELAY COMPETITION ROUND 3 ICTM 2010 DIVISION AA STATE FINALS QUESTION 1 1. A book contains exactly 67 pages numbered from 1—67 inclusive. How many

times was the digit 3 used in this numbering?

FRESHMAN-SOPHOMORE RELAY COMPETITION ROUND 3 ICTM 2010 DIVISION AA STATE FINALS QUESTION 2 2. Let { }1, 2,3, 4,5,6,7,8,9,10A = . One of the ten numbers in A is selected at

random. Find the probability that the number selected is a member of the solution set for the inequality ( ) ( )5 9 3 11x x ANS+ − + > . Express your answer as a common fraction reduced to lowest terms.

FRESHMAN-SOPHOMORE RELAY COMPETITION ROUND 3 ICTM 2010 DIVISION AA STATE FINALS QUESTION 3

3. ANS should be a common fraction in the form kw

where k and w are relatively

prime integers. Let f k w= + . Three coplanar circles are placed so that each circle is tangent externally to each of the other two circles. The respective centers of these circles are point A , B , and C . The radius of the smallest circle is 2. If AB f= and 33BC = , find the smallest possible value for the length of AC .

FRESHMAN-SOPHOMORE RELAY COMPETITION ROUND 3 ICTM 2010 DIVISION AA STATE FINALS QUESTION 4 4. The length of one of the legs of a right triangle is ANS . The length of each side

of the right triangle is a positive integer. The length of the other leg of the right triangle is k . If 50k < , find the sum of all distinct possible values of k .

FRESHMAN-SOPHOMORE RELAY COMPETITION ROUND 4 ICTM 2010 DIVISION AA STATE FINALS QUESTION 1 1. Edward starts at point A and walks due north at 3 mph. Two hours later, Ville

starts at point A and walks due north at 4 mph. Find the number of minutes that Ville will walk before she catches Edward.

FRESHMAN-SOPHOMORE RELAY COMPETITION ROUND 4 ICTM 2010 DIVISION AA STATE FINALS QUESTION 2 2. Five times a number added to twenty more than twelve times the number is ANS .

Find the number.

FRESHMAN-SOPHOMORE RELAY COMPETITION ROUND 4 ICTM 2010 DIVISION AA STATE FINALS QUESTION 3 3. The lengths of the diagonals of a rhombus are respectively ANS and 48. Find the

area of the rhombus.

FRESHMAN-SOPHOMORE RELAY COMPETITION ROUND 4 ICTM 2010 DIVISION AA STATE FINALS QUESTION 4 4. The perimeter of a regular hexagon is ANS . Find the positive difference between

the radius of a circumscribed circle of this hexagon and the radius of an inscribed circle of this hexagon. Express your answer as a decimal rounded to the nearest hundredth.

FRESHMAN-SOPHOMORE RELAY COMPETITION ROUND 5 ICTM 2010 DIVISION AA STATE FINALS QUESTION 1

2 3 4 59 8 7 6

10 11 12 1317 16 15 14

1. Given the 5 column arrangement of integers as shown.

In the first row, 2 falls in column 2, 3 falls in column 3, 4 falls in column 4, and 5 falls in column 5. If the pattern continues, find the number of the column in which 37 will fall.

FRESHMAN-SOPHOMORE RELAY COMPETITION ROUND 5 ICTM 2010 DIVISION AA STATE FINALS QUESTION 2

2. If ( 11)1 3216

ANSx−⎛ ⎞ =⎜ ⎟

⎝ ⎠, find the value of 2x .

FRESHMAN-SOPHOMORE RELAY COMPETITION ROUND 5 ICTM 2010 DIVISION AA STATE FINALS QUESTION 3 3. In the figure, points A , B , D , and E lie on the circle. Points A , B , and C are

collinear, and points C , D , and E are collinear. If ( )AE ANS= ° and

104BD = ° , find the number of degrees in ACE∠ .

D

B

CE

A

FRESHMAN-SOPHOMORE RELAY COMPETITION ROUND 5 ICTM 2010 DIVISION AA STATE FINALS QUESTION 4 4. Two coplanar circles are concentric. A chord of the larger circle is tangent to the

smaller circle. The length of that chord is ANS . A radius of the larger circle has a length of 10. The area of the region bounded between the two circles is kπ . Find the value of k .

2 39 8 7

10 1117 16 15

2010 SAA FR/SO RELAY COMPETITION

PROCTOR ANSWER SHEET ROUND 1 1. 5.9 (Must be this decimal.) 2. 32.6 (Must be this decimal, minutes

optional.) 3. 76.5 (Must be this decimal, degrees

optional.) 4. 74001.2 (Must be this decimal.) ROUND 2 1. 34650 2. 3 3. 2.598 (Must be this decimal.) 4. 9.526 (Must be this decimal.) ROUND 3 1. 17

2. 45

(Must be this reduced common

fraction.) 3. 28 4. 66

EXTRA ROUND 4 1. 360 (Minutes optional.) 2. 20 3. 480 4. 10.72 (Must be this decimal.) EXTRA ROUND 5 1. 5 2. 128 3. 12 (Degrees optional.) 4. 36

JUNIOR-SENIOR RELAY COMPETITION ROUND 1 ICTM 2010 DIVISION AA STATE FINALS QUESTION 1 1. Let 9k x= − + 5089. If 0 60x< < , find the sum of all distinct values of x such that k

is a positive integer.

JUNIOR-SENIOR RELAY COMPETITION ROUND 1 ICTM 2010 DIVISION AA STATE FINALS QUESTION 2 2. By how much does the sum of the integers from 205 through ANS inclusive exceed the

sum of the integers from 205 through 210 inclusive?

JUNIOR-SENIOR RELAY COMPETITION ROUND 1 ICTM 2010 DIVISION AA STATE FINALS QUESTION 3 3. Let 105k ANS= − . You possess a certain number of pennies and quarters, and their total

value is k cents. If you had a third as many quarters and three times as many pennies, the total value would be five cents less than half of k cents. Find the total number of coins that you possess that are pennies or quarters.

JUNIOR-SENIOR RELAY COMPETITION ROUND 1 ICTM 2010 DIVISION AA STATE FINALS QUESTION 4 4. If 43.1x > , find the value of

3 2

2

62 1135 5250log( ) log( 30) 28.5232 175

x x x x ANSx x

− + −− − + −

− +.

Express your answer as a decimal rounded to the nearest hundredth.

JUNIOR-SENIOR RELAY COMPETITION ROUND 2 ICTM 2010 DIVISION AA STATE FINALS QUESTION 1 1. Find the maximum number of internal regions into which 3 lines can divide a circle.

JUNIOR-SENIOR RELAY COMPETITION ROUND 2 ICTM 2010 DIVISION AA STATE FINALS QUESTION 2 2. All diagonals of a regular hexagon are drawn. The hexagon is now divided internally

into a maximum number of non-overlapping (disjoint) internal polygonal regions. If ANS of these disjoint internal polygonal regions are shaded, and the other disjoint internal polygonal regions are unshaded, find the number of these disjoint internal polygonal regions that are unshaded.

JUNIOR-SENIOR RELAY COMPETITION ROUND 2 ICTM 2010 DIVISION AA STATE FINALS QUESTION 3 3. Let k ANS= . Tom has a k per cent chance of winning 0 promprins, a ( )2k per cent

chance of winning 10 promprins; otherwise, he wins 86049

promprins. Find the number of

promprins in his mathematical expectation.

JUNIOR-SENIOR RELAY COMPETITION ROUND 2 ICTM 2010 DIVISION AA STATE FINALS QUESTION 4

4. An ellipse has an equation of 2 2

2

( 1) ( 1) 1( ) 121x yANS− +

+ = . One of the foci of this

ellipse has coordinates (1 , 1)k+ − where k is a positive integer. Find the value of k .

JUNIOR-SENIOR RELAY COMPETITION ROUND 3 ICTM 2010 DIVISION AA STATE FINALS QUESTION 1 1. Adlai is now twice as old as his sister Christina was eight years ago. Find the number of

years in Adlai’s age twelve years from now if Christina then will be 21 years old.

JUNIOR-SENIOR RELAY COMPETITION ROUND 3 ICTM 2010 DIVISION AA STATE FINALS QUESTION 2 2. A total of ANS distinct diagonals can be drawn in a certain convex polygon. Find the

number of sides of this polygon.

JUNIOR-SENIOR RELAY COMPETITION ROUND 3 ICTM 2010 DIVISION AA STATE FINALS QUESTION 3 3. There are exactly ANS distinct marbles in a bag—each of which is a different color.

Karen picks two of these marbles at random (with replacement). If Karen names one of the colors of the marbles before drawing, find the probability that she picked two marbles of the same color but that the two marbles picked were not of the color that Karen named before drawing. Express your answer as a common fraction reduced to lowest terms.

JUNIOR-SENIOR RELAY COMPETITION ROUND 3 ICTM 2010 DIVISION AA STATE FINALS QUESTION 4

4. ANS should be a common fraction of the form kw

where k and w are relatively prime

positive integers. Let p be a positive integer such that 0 p k< ≤ . Find the sum of all

distinct possible values of log( )kpw

. Express your answer as a decimal rounded to

the nearest thousandth.

JUNIOR-SENIOR RELAY COMPETITION ROUND 4 ICTM 2010 DIVISION AA STATE FINALS QUESTION 1 1. In right triangle DAC, DAC∠ is a right angle. Point B

lies on AC as shown. If DA = 8, CD = 17, and DB = 10, find BC .

D

A CB

JUNIOR-SENIOR RELAY COMPETITION ROUND 4 ICTM 2010 DIVISION AA STATE FINALS QUESTION 2 2. ANS is the length of a leg in each of two right triangles. The hypotenuse of one right

triangle has a length of 41, and the hypotenuse of the other right triangle has a length of 15. By how much does the length of the remaining leg of the larger right triangle exceed the length of the remaining leg of the smaller right triangle?

JUNIOR-SENIOR RELAY COMPETITION ROUND 4 ICTM 2010 DIVISION AA STATE FINALS QUESTION 3 3. ANS is the sum of the terms of an infinite geometric sequence whose first term is 20.

Find the ratio of the fourth term to the third term. Express your answer as a common fraction reduced to lowest terms.

JUNIOR-SENIOR RELAY COMPETITION ROUND 4 ICTM 2010 DIVISION AA STATE FINALS QUESTION 4 4. The cosine of the largest acute angle of a right triangle is ANS . Find the sum of the

sines of the other two angles of this right triangle. Express your answer as an improper fraction reduced to lowest terms.

JUNIOR-SENIOR RELAY COMPETITION ROUND 5 ICTM 2010 DIVISION AA STATE FINALS QUESTION 1 1. Let ( ) 17 8f x x= + . By how much does ( )3 2f exceed ( )2 3f ?

JUNIOR-SENIOR RELAY COMPETITION ROUND 5 ICTM 2010 DIVISION AA STATE FINALS QUESTION 2 2. Let k , w , and p represent integers. The quartic equation 4 3 2 144 0x kx wx px+ + + − =

factors into 2 2( ( ) 48)( 3) 0x ANS x x x+ − + + = . One of the roots of that quartic equation is a positive integer. That integer is the length of a side of an equilateral triangle. Find the area of that equilateral triangle.

JUNIOR-SENIOR RELAY COMPETITION ROUND 5 ICTM 2010 DIVISION AA STATE FINALS QUESTION 3 3. The points ( 45 , 8 3 )− and (3 5 , )ANS are the endpoints of a diameter of a circle.

Find the y-coordinate of the center of this circle.

JUNIOR-SENIOR RELAY COMPETITION ROUND 5 ICTM 2010 DIVISION AA STATE FINALS QUESTION 4 4. When simplified, ANS should be in the form k w . Find the real value of x which is a

solution of the equation 1 11 1 0kwx x

⎛ ⎞+ + + + =⎜ ⎟⎝ ⎠

.

2010 SAA JR/SR RELAY COMPETITION

PROCTOR ANSWER SHEET ROUND 1 1. 212 2. 423 3. 30 4. 1.48 (Must be this decimal.) ROUND 2 1. 7 2. 17 3. 12 (Promprins optional.) 4. 23 ROUND 3 1. 14 (Years optional.) 2. 7

3. 649

(Must be this reduced common

fraction.) 4. 2.615− (Must be this decimal.)

EXTRA ROUND 4 1. 9 2. 28

3. 27

(Must be this reduced common

fraction.)

4. 97

(Must be this reduced improper

fraction.) EXTRA ROUND 5 1. 8 2. 4 3 3. 2 3−

4. 13

or .3 or 0.3

Questions for the Oral Competition – Division AA, State Finals 2010 1) Using only polygons with 15 or fewer sides, can a semiregular tiling exist whose vertex figure

contains exactly one octagon? 2) What combination(s) of regular shapes that contain at least one regular hexagon can be

used to tile a plane? 3) Does the set of odd integers form a group under the operation of addition? 4) Consider the regular hexagon ABCDEF and the rigid motion of a - a reflection H across a horizontal line of symmetry of the hexagon - a reflection V across a vertical line of symmetry of the hexagon - a rotation R of 60° about the center of the hexagon

Explain why any combination of these rigid motions cannot result in the hexagon shown below:

5) Below is a drawing of a Penrose tile. If the length of DE is 2 units, give the exact lengths of the segments BE, AE, CE, AB, AD, DC and BC .

A B

C

D E

F

F E

D

C A

B

Questions for the Oral Competition – Division AA, State Finals 2010

SOLUTIONS 1) Using only polygons with 15 or fewer sides, can a semiregular tiling exist whose vertex figure

contains exactly one octagon?

No. One octagon has an interior angle of 135° and no other combination of regular figures with less than 20 sides have interior angles which would allow the angles around each vertex to add to 360°

2) What combination(s) of regular shapes that contain at least one regular hexagon can be

used to tile a plane? 3 hexagons 2 hexagon, 2 triangles 1 hexagon, 4 triangles 1 hexagon, 1 triangle, 2 squares 3) Does the set of odd integers form a group under the operation of addition? No. Closure does not hold (odd integer + odd integer = even integer)

4) Consider the regular hexagon ABCDEF and the rigid motions of a - a reflection H across a horizontal line of symmetry of the hexagon - a reflection V across a vertical line of symmetry of the hexagon - a rotation R of 60° about the center of the hexagon

Explain why any combination of these rigid motions cannot result in the hexagon shown below:

All of the rigid motions maintain size, shape, and order of the vertices. Since the vertices of the resulting figure (BACDEF) are not in the same order as the original figure (ABCDEF), it is not possible.

5) Below is a drawing of a Penrose tile. If the length of DE is 2 units, give the exact lengths of the segments BE, AE, CE, AB, AD, DC and BC .

A B

C

D E

F

F E

D

C A

B

72° 36° 36°

36°

72° 36°

36°

72° 72°

36°

The ratio of the segments of the diagonal in a Penrose tile is 1:φ (shorter segment: longer segment). If DE=2, then BE= 2φ . With angles as shown below, BE=BC=AB and DE=EC=AE. Since the Penrose tile is a rhombus, AD=CD=AB=BC .

DE = EC = AE = 2 BE = AB = AD = CD = BC = 2φ

Extemporaneous Questions for the Oral Competition – Division AA, State Finals 2010

1) Which regular polygons can be used to create a monhedral tiling?

2) The parallelogram ABCD (with dashed lines) is being adapted as shown by the solid lines in

the second figure to be used to tile the plane. Describe how segments AB and AD in the second figure need to be changed to be able to do so.

A

B C

D A

B C

D

3) Geometric transformations also occur in musical competitions. For each piece of “music” shown below on a unique ICTM music staff with unique “notes,” state whether the second figure illustrates a reflection, rotation, translation, or glide reflection of the first. Explain each answer. a) b) c) 4) If a glide reflection is performed on the following group of “notes,” sketch the resulting figure.

Extemporaneous Questions for the Oral Competition – Division AA, State Finals 2010

SOLUTIONS

1) Which regular polygons can be used to create a monhedral tiling?

Triangles, squares and hexagons are the only regular figures that can create a monhedral tiling.

2) The parallelogram ABCD (with dashed lines) is being adapted as shown by the solid lines in

the second figure to be used to tile the plane. Describe how segments AB and AD in the second figure need to be changed to be able to do so.

Solution:

B C

D A

A

B C

D A

B C

D

3) Geometric transformations also occur in musical competitions. For each piece of “music” shown below on a unique ICTM music “staff” with unique “notes,” state whether the second figure illustrates a reflection, rotation, translation, or glide reflection of the first. Explain each answer. a) a) reflection – the second set of notes has been reflected over the middle line in the staff. b) b) translation – the second set of notes has been shifted to the right along a horizontal line c) c) rotation – the second set of notes has been rotated 90 degrees to the right. 4) Perform a glide reflection on the following group of “notes” and sketch the resulting figure. One possibility is shown here – credit should be given for any valid glide reflection.