people.math.sc.edu · Remarks on Writing a High School Competition: Original Problem 23: When Tammy...

HIGH SCHOOL M ATH CONTEST

UNIVERSITY OF SOUTH CAROLINA

JANUARY 19, 2002

Problem 4:

Average for women: 83Average for men: 71Average for all: 80

What percentage of the students are women?

Problem 4:

Number of women:w

Problem 4:

Number of women:w

Number of students:N

Problem 4:

Number of women:wNumber of men:N − wNumber of students:N

Problem 4:

sum of the scores in the class

Problem 4:

= sum of women scores+ sum of men scores

sum of the scores in the class

Problem 4:

80 · N = sum of the scores in the class

Problem 4:

︸︷︷︸

Problem 4:

= sum of women scores︸︷︷︸

83·w+ sum of men scores︸︷︷︸

71·(N−w)

Problem 4:

80N = 83w + 71(N − w)

Problem 4:

80N = 83w + 71(N − w) =⇒ 9N = 12w

Problem 4:

80N = 83w + 71(N − w) =⇒ 9N = 12w

Problem 4:

80N = 83w + 71(N − w) =⇒ 9N = 12w

Problem 4:

80N = 83w + 71(N − w) =⇒ 9N = 12w

12= 75%

Problem 4:

80N = 83w + 71(N − w) =⇒ 9N = 12w

12= 75%

Problem 5:

3a = 4, 4b = 5, 5c = 6

6d = 7, 7e = 8, 8f = 9

Calculateabcdef?

Problem 5:

3a = 4, 4b = 5, 5c = 6

6d = 7, 7e = 8, 8f = 9

Calculateabcdef?

3abcdef

Problem 5:

3a = 4, 4b = 5, 5c = 6

6d = 7, 7e = 8, 8f = 9

Calculateabcdef?

3abcdef =(3a

)bcdef

Problem 5:

3a = 4, 4b = 5, 5c = 6

6d = 7, 7e = 8, 8f = 9

Calculateabcdef?

3abcdef =(3a

)bcdef

Problem 5:

3a = 4, 4b = 5, 5c = 6

6d = 7, 7e = 8, 8f = 9

Calculateabcdef?

3abcdef =(4

)bcdef

Problem 5:

3a = 4, 4b = 5, 5c = 6

6d = 7, 7e = 8, 8f = 9

Calculateabcdef?

3abcdef =(4

)bcdef

Problem 5:

3a = 4, 4b = 5, 5c = 6

6d = 7, 7e = 8, 8f = 9

Calculateabcdef?

3abcdef =(4

)bcdef

Problem 5:

3a = 4, 4b = 5, 5c = 6

6d = 7, 7e = 8, 8f = 9

Calculateabcdef?

3abcdef =(4

)bcdef

Problem 5:

3a = 4, 4b = 5, 5c = 6

6d = 7, 7e = 8, 8f = 9

Calculateabcdef?

3abcdef =(4

)bcdef

Problem 5:

3a = 4, 4b = 5, 5c = 6

6d = 7, 7e = 8, 8f = 9

Calculateabcdef?

3abcdef =(4

)bcdef

= 6def

Problem 5:

3a = 4, 4b = 5, 5c = 6

6d = 7, 7e = 8, 8f = 9

Calculateabcdef?

3abcdef =(4

)bcdef

Problem 5:

3a = 4, 4b = 5, 5c = 6

6d = 7, 7e = 8, 8f = 9

Calculateabcdef?

3abcdef =(4

)bcdef

Problem 5:

3a = 4, 4b = 5, 5c = 6

6d = 7, 7e = 8, 8f = 9

Calculateabcdef?

3abcdef =(4

)bcdef

Problem 5:

3a = 4, 4b = 5, 5c = 6

6d = 7, 7e = 8, 8f = 9

Calculateabcdef?

3abcdef =(4

)bcdef

Problem 5:

3a = 4, 4b = 5, 5c = 6

6d = 7, 7e = 8, 8f = 9

Calculateabcdef?

3abcdef =(4

)bcdef

abcdef = 2

Problem 5:

3a = 4, 4b = 5, 5c = 6

6d = 7, 7e = 8, 8f = 9

Calculateabcdef?

3abcdef =(4

)bcdef

abcdef = 2

Problem 7:

PROBLEM 7:

30 multiple choice questions

PROBLEM 7:

5 points for a correct answer

PROBLEM 7:

1 point for no answer

PROBLEM 7:

0 points for a wrong answer

PROBLEM 7:

Which of the scores 147, 144, 143, 141, 139 is possible?

PROBLEM 7:

Which of the scores 147, 144, 143, 141, 139 is possible?

x problems correct,y with no answer

=⇒ 0 ≤ x + y ≤ 30 and the score is5x + y

PROBLEM 7:

For eachS ∈ {147, 144, 143, 141, 139}, are there non-negative integersx andy such that

0 ≤ x + y ≤ 30 and S = 5x + y ?

PROBLEM 7:

0 ≤ x + y ≤ 30 and S = 5x + y ?

141 = 5 × 28 + 1

PROBLEM 7:

0 ≤ x + y ≤ 30 and S = 5x + y ?

141 = 5 × 28 + 1 =⇒ the answer is141

PROBLEM 7:

0 ≤ x + y ≤ 30 and S = 5x + y ?

141 = 5 × 28 + 1 =⇒ the answer is141

Is this the only correct answer?

PROBLEM 7:

0 ≤ x + y ≤ 30 and S = 5x + y ?

S = 5x + y

PROBLEM 7:

0 ≤ x + y ≤ 30 and S = 5x + y ?

S = 5x + y = 5(x + y) − 4y

PROBLEM 7:

0 ≤ x + y ≤ 30 and S = 5x + y ?

S = 5x + y = 5(x + y) − 4y

PROBLEM 7:

0 ≤ x + y ≤ 30 and S = 5x + y ?

S = 5x + y = 5(x + y) − 4y ≤ 150 − 4y

PROBLEM 7:

0 ≤ x + y ≤ 30 and S = 5x + y ?

S = 5x + y = 5(x + y) − 4y ≤ 150 − 4y

PROBLEM 7:

0 ≤ x + y ≤ 30 and S = 5x + y ?

S = 5x + y = 5(x + y) − 4y ≤ 150 − 4y

S = 5x + y

PROBLEM 7:

0 ≤ x + y ≤ 30 and S = 5x + y ?

S = 5x + y = 5(x + y) − 4y ≤ 150 − 4y

S = 5x + y ≤ 150 − 4 · 2 = 142

PROBLEM 7:

0 ≤ x + y ≤ 30 and S = 5x + y ?

S = 5x + y = 5(x + y) − 4y ≤ 150 − 4y

S = 5x + y

PROBLEM 7:

0 ≤ x + y ≤ 30 and S = 5x + y ?

S = 5x + y = 5(x + y) − 4y ≤ 150 − 4y

S = 5x + y ≤ 150 − 4 · 3 = 138

What’s the smallest score thatis not obtainable on this test?

The only nonobtainable scores are:

139, 143

139, 143, 144

139, 143, 144, 147

139, 143, 144, 147, 148, 149

and the obvious ones like:

151, −28,√

2,√−1,

e − 17

Problem 10:

a, b, andc are different

Problem 10:

Main Idea:

Problem 10:

Main Idea: Add 1 to each expression.

Problem 10:

1 +a + b

Problem 10:

1 +a + b

a + b + c

Problem 10:

1 +a + b

a + b + c

1 +b + c

Problem 10:

1 +a + b

a + b + c

1 +b + c

a + b + c

Problem 10:

1 +a + b

a + b + c

1 +b + c

a + b + c

1 +c + a

Problem 10:

1 +a + b

a + b + c

1 +b + c

a + b + c

1 +c + a

a + b + c

Problem 10:

a + b + c

Problem 10:

a = b = c

Problem 10:

a + b + c

Problem 10:

a + b + c

a + b + c = 0

Problem 10:

a + b + c

a + b + c = 0

Problem 10:

a + b + c

a + b + c = 0

c= −1

Problem 10:

a + b + c

a + b + c = 0

c= −1

Problem 10:

Takea = 1, b = 2, andc = −3.

Problem 18:

a + b + c + d = 10 (a + b)(c + d) = 16

(a + c)(b + d) = 21 (a + d)(b + c) = 24

Problem 18:

a + b + c + d = 10 (a + b)(c + d) = 16

(a + c)(b + d) = 21 (a + d)(b + c) = 24

a2 + b2 + c2 + d2 = ?

Problem 18:

a + b + c + d = 10 (a + b)(c + d) = 16

(a + c)(b + d) = 21 (a + d)(b + c) = 24

a2 + b2 + c2 + d2 = ?

a2+b2 + c2 + d2

= (a + b + c + d)2 − (a + b)(c + d)

− (a + c)(b + d) − (a + d)(b + c)

Problem 18:

a + b + c + d = 10 (a + b)(c + d) = 16

(a + c)(b + d) = 21 (a + d)(b + c) = 24

a2 + b2 + c2 + d2 = ?

a2+b2 + c2 + d2

= (a + b + c + d)2 − (a + b)(c + d)

− (a + c)(b + d) − (a + d)(b + c)

= 102 − 16 − 21 − 24 = 39

Problem 18:

a + b + c + d = 10 (a + b)(c + d) = 16

(a + c)(b + d) = 21 (a + d)(b + c) = 24

a2 + b2 + c2 + d2 = ?

a2+b2 + c2 + d2

= (a + b + c + d)2 − (a + b)(c + d)

− (a + c)(b + d) − (a + d)(b + c)

= 102 − 16 − 21 − 24 = 39

Problem 18:

a2+b2 + c2 + d2

= (a + b + c + d)2 − (a + b)(c + d)

− (a + c)(b + d) − (a + d)(b + c)

Problem 18:

a2+b2 + c2 + d2

= (a + b + c + d)2 − (a + b)(c + d)

− (a + c)(b + d) − (a + d)(b + c)

(a + b + c + d)2

= a2 + b2 + c2 + d2 + (a + b)(c + d)

+ (a + c)(b + d) + (a + d)(b + c)

Problem 18:

a2+b2 + c2 + d2

= (a + b + c + d)2 − (a + b)(c + d)

− (a + c)(b + d) − (a + d)(b + c)

(a + b + c + d)(a + b + c + d)

= a2 + b2 + c2 + d2 + (a + b)(c + d)

+ (a + c)(b + d) + (a + d)(b + c)

Problem 18:

a2+b2 + c2 + d2

= (a + b + c + d)2 − (a + b)(c + d)

− (a + c)(b + d) − (a + d)(b + c)

(a + b + c + d)(a + b + c + d)

= a2 + b2 + c2 + d2 + (a + b)(c + d)

+ (a + c)(b + d) + (a + d)(b + c)

Problem 18:

a2+b2 + c2 + d2

= (a + b + c + d)2 − (a + b)(c + d)

− (a + c)(b + d) − (a + d)(b + c)

(a + b + c + d)(a + b + c + d)

= a2 + b2 + c2 + d2 + (a + b)(c + d)

+ (a + c)(b + d) + (a + d)(b + c)

Problem 18:

a2+b2 + c2 + d2

= (a + b + c + d)2 − (a + b)(c + d)

− (a + c)(b + d) − (a + d)(b + c)

(a + b + c + d)(a + b + c + d)

= a2 + b2 + c2 + d2 + (a + b)(c + d)

+ (a + c)(b + d) + (a + d)(b + c)

Problem 18:

a2+b2 + c2 + d2

= (a + b + c + d)2 − (a + b)(c + d)

− (a + c)(b + d) − (a + d)(b + c)

(a + b + c + d)(a + b + c + d)

= a2 + b2 + c2 + d2 + (a + b)(c + d)

+ (a + c)(b + d) + (a + d)(b + c)

Problem 18:

a2+b2 + c2 + d2

= (a + b + c + d)2 − (a + b)(c + d)

− (a + c)(b + d) − (a + d)(b + c)

(a + b + c + d)(a + b + c + d)

= a2 + b2 + c2 + d2 + (a + b)(c + d)

+ (a + c)(b + d) + (a + d)(b + c)

Problem 18:

a2+b2 + c2 + d2

= (a + b + c + d)2 − (a + b)(c + d)

− (a + c)(b + d) − (a + d)(b + c)

(a + b + c + d)(a + b + c + d)

= a2 + b2 + c2 + d2 + (a + b)(c + d)

+ (a + c)(b + d) + (a + d)(b + c)

Problem 18:

a2+b2 + c2 + d2

= (a + b + c + d)2 − (a + b)(c + d)

− (a + c)(b + d) − (a + d)(b + c)

(a + b + c + d)(a + b + c + d)

= a2 + b2 + c2 + d2 + (a + b)(c + d)

+ (a + c)(b + d) + (a + d)(b + c)

Problem 18:

a2+b2 + c2 + d2

= (a + b + c + d)2 − (a + b)(c + d)

− (a + c)(b + d) − (a + d)(b + c)

(a + b + c + d)(a + b + c + d)

= a2 + b2 + c2 + d2 + (a + b)(c + d)

+ (a + c)(b + d) + (a + d)(b + c)

Problem 18:

a2+b2 + c2 + d2

= (a + b + c + d)2 − (a + b)(c + d)

− (a + c)(b + d) − (a + d)(b + c)

(a + b + c + d)(a + b + c + d)

= a2 + b2 + c2 + d2 + (a + b)(c + d)

+ (a + c)(b + d) + (a + d)(b + c)

Problem 18:

a2+b2 + c2 + d2

= (a + b + c + d)2 − (a + b)(c + d)

− (a + c)(b + d) − (a + d)(b + c)

(a + b + c + d)(a + b + c + d)

= a2 + b2 + c2 + d2 + (a + b)(c + d)

+ (a + c)(b + d) + (a + d)(b + c)

Problem 18:

a2+b2 + c2 + d2

= (a + b + c + d)2 − (a + b)(c + d)

− (a + c)(b + d) − (a + d)(b + c)

(a + b + c + d)(a + b + c + d)

= a2 + b2 + c2 + d2 + (a + b)(c + d)

+ (a + c)(b + d) + (a + d)(b + c)

Problem 18:

a + b + c + d = 10 (a + b)(c + d) = 16

(a + c)(b + d) = 21 (a + d)(b + c) = 24

a2 + b2 + c2 + d2 = ?

Problem 18:

a + b + c + d = 10 (a + b)(c + d) = 16

(a + c)(b + d) = 21 (a + d)(b + c) = 24

a2 + b2 + c2 + d2 = ?

(x − (a + b)

)(x − (c + d)

Problem 18:

a + b + c + d = 10 (a + b)(c + d) = 16

(a + c)(b + d) = 21 (a + d)(b + c) = 24

a2 + b2 + c2 + d2 = ?

(x − (a + b)

)(x − (c + d)

= x2 − (a + b + c + d)x + (a + b)(c + d)

Problem 18:

a + b + c + d = 10 (a + b)(c + d) = 16

(a + c)(b + d) = 21 (a + d)(b + c) = 24

a2 + b2 + c2 + d2 = ?

(x − (a + b)

)(x − (c + d)

= x2 − (a + b + c + d)x + (a + b)(c + d)

= x2 − 10x + 16

Problem 18:

a + b + c + d = 10 (a + b)(c + d) = 16

(a + c)(b + d) = 21 (a + d)(b + c) = 24

a2 + b2 + c2 + d2 = ?

(x − (a + b)

)(x − (c + d)

= x2 − (a + b + c + d)x + (a + b)(c + d)

= x2 − 10x + 16

= (x − 2)(x − 8)

Problem 18:

a + b + c + d = 10 a + b = 2

(a + c)(b + d) = 21 (a + d)(b + c) = 24

a2 + b2 + c2 + d2 = ?

(x − (a + b)

)(x − (c + d)

= x2 − (a + b + c + d)x + (a + b)(c + d)

= x2 − 10x + 16

= (x − 2)(x − 8)

Problem 18:

a + b + c + d = 10 a + b = 2

(a + c)(b + d) = 21 (a + d)(b + c) = 24

a2 + b2 + c2 + d2 = ?

(x − (a + c)

)(x − (b + d)

= x2 − (a + b + c + d)x + (a + c)(b + d)

= x2 − 10x + 21

= (x − 3)(x − 7)

Problem 18:

a + b + c + d = 10 a + b = 2

a + c = 3 (a + d)(b + c) = 24

a2 + b2 + c2 + d2 = ?

(x − (a + c)

)(x − (b + d)

= x2 − (a + b + c + d)x + (a + c)(b + d)

= x2 − 10x + 21

= (x − 3)(x − 7)

Problem 18:

a + b + c + d = 10 a + b = 2

a + c = 3 (a + d)(b + c) = 24

a2 + b2 + c2 + d2 = ?

(x − (a + d)

)(x − (b + c)

= x2 − (a + b + c + d)x + (a + d)(b + c)

= x2 − 10x + 24

= (x − 4)(x − 6)

Problem 18:

a + b + c + d = 10 a + b = 2

a + c = 3 a + d = 4

a2 + b2 + c2 + d2 = ?

(x − (a + d)

)(x − (b + c)

= x2 − (a + b + c + d)x + (a + d)(b + c)

= x2 − 10x + 24

= (x − 4)(x − 6)

Problem 18:

a + b + c + d = 10 a + b = 2

a + c = 3 a + d = 4

a2 + b2 + c2 + d2 = ?

Problem 18:

a + b + c + d = 10 a + b = 2

a + c = 3 a + d = 4

a2 + b2 + c2 + d2 = ?

2a = (a + b)+(a + c) + (a + d)

− (a + b + c + d)

= −1

Problem 18:

a + b + c + d = 10 a + b = 2

a + c = 3 a + d = 4

a2 + b2 + c2 + d2 = ?

2a = (a + b)+(a + c) + (a + d)

− (a + b + c + d)

= −1

a = −1/2

Problem 18:

a + b + c + d = 10 a + b = 2

a + c = 3 a + d = 4

a2 + b2 + c2 + d2 = ?

2a = (a + b)+(a + c) + (a + d)

− (a + b + c + d)

= −1

a = −1/2, b = 5/2

Problem 18:

a + b + c + d = 10 a + b = 2

a + c = 3 a + d = 4

a2 + b2 + c2 + d2 = ?

2a = (a + b)+(a + c) + (a + d)

− (a + b + c + d)

= −1

a = −1/2, b = 5/2, c = 7/2, d = 9/2

Problem 19:

[x] = greatest integer≤ x

Problem 19:

bxc = greatest integer≤ x

Problem 19:

4 ≤ x < 5 =⇒ [x] = 4

Problem 19:

a · [a] = 17, b · [b] = 11, a − b = ?

Problem 19:

a · [a] = 17, b · [b] = 11, a − b = ?

4 ≤ a

Problem 19:

a · [a] = 17, b · [b] = 11, a − b = ?

4 ≤ a < 5

Problem 19:

a · [a] = 17, b · [b] = 11, a − b = ?

4 ≤ a < 5 =⇒ [a] = 4

Problem 19:

a · [a] = 17, b · [b] = 11, a − b = ?

4 ≤ a < 5 =⇒ [a] = 4

Problem 19:

a · [a] = 17, b · [b] = 11, a − b = ?

4 ≤ a < 5 =⇒ [a] = 4 =⇒ a =17

Problem 19:

a · [a] = 17, b · [b] = 11, a − b = ?

4 ≤ a < 5 =⇒ [a] = 4 =⇒ a =17

3 ≤ b < 4

Problem 19:

a · [a] = 17, b · [b] = 11, a − b = ?

4 ≤ a < 5 =⇒ [a] = 4 =⇒ a =17

3 ≤ b < 4 =⇒ [b] = 3

Problem 19:

a · [a] = 17, b · [b] = 11, a − b = ?

4 ≤ a < 5 =⇒ [a] = 4 =⇒ a =17

3 ≤ b < 4 =⇒ [b] = 3 =⇒ b =11

Problem 19:

a · [a] = 17, b · [b] = 11, a − b = ?

4 ≤ a < 5 =⇒ [a] = 4 =⇒ a =17

3 ≤ b < 4 =⇒ [b] = 3 =⇒ b =11

a − b =17

4− 11

Problem 19:

a · [a] = 17, b · [b] = 11, a − b = ?

4 ≤ a < 5 =⇒ [a] = 4 =⇒ a =17

3 ≤ b < 4 =⇒ [b] = 3 =⇒ b =11

a − b =17

4− 11

51 − 44

Problem 19:

a · [a] = 17, b · [b] = 11, a − b = ?

4 ≤ a < 5 =⇒ [a] = 4 =⇒ a =17

3 ≤ b < 4 =⇒ [b] = 3 =⇒ b =11

a − b =17

4− 11

Problem 19:

a · [a] = 17, b · [b] = 11, a − b = ?

4 ≤ a < 5 =⇒ [a] = 4 =⇒ a =17

3 ≤ b < 4 =⇒ [b] = 3 =⇒ b =11

a − b =17

4− 11

Problem 22:

1000 · 1000! + 999 · 999! + · · · + 2 · 2! + 1 · 1! = ?

Problem 22:

1000 · 1000! + 999 · 999! + · · · + 2 · 2! + 1 · 1! = ?

Main Idea:

Problem 22:

1000 · 1000! + 999 · 999! + · · · + 2 · 2! + 1 · 1! = ?

Main Idea: (n + 1)! = (n + 1) · n!

Problem 22:

1000 · 1000! + 999 · 999! + · · · + 2 · 2! + 1 · 1! = ?

Main Idea: (n + 1)! = (n + 1) · n! = n · n! + n!

Problem 22:

1000 · 1000! + 999 · 999! + · · · + 2 · 2! + 1 · 1! = ?

Main Idea: (n + 1)! = (n + 1) · n! = n · n! + n!

Problem 22:

1000 · 1000! + 999 · 999! + · · · + 2 · 2! + 1 · 1! = ?

Main Idea: (n + 1)! = n · n! + n!

Problem 22:

1000 · 1000! + 999 · 999! + · · · + 2 · 2! + 1 · 1! = ?

Main Idea: (n + 1)! = n · n! + n!

1001! = 1000 · 1000! + 1000!

Problem 22:

1000 · 1000! + 999 · 999! + · · · + 2 · 2! + 1 · 1! = ?

Main Idea: (n + 1)! = n · n! + n!

1001! = 1000 · 1000! + 1000!

= 1000 · 1000! + 999 · 999! + 999!

Problem 22:

1000 · 1000! + 999 · 999! + · · · + 2 · 2! + 1 · 1! = ?

Main Idea: (n + 1)! = n · n! + n!

1001! = 1000 · 1000! + 1000!

= 1000 · 1000! + 999 · 999! + 999!

= 1000 · 1000! + 999 · 999! + 998 · 998! + 998!

Problem 22:

1000 · 1000! + 999 · 999! + · · · + 2 · 2! + 1 · 1! = ?

Main Idea: (n + 1)! = n · n! + n!

1001! = 1000 · 1000! + 1000!

= 1000 · 1000! + 999 · 999! + 999!

= 1000 · 1000! + 999 · 999! + 998 · 998! + 998!

= · · · = 1000 · 1000! + · · · + 2 · 2! + 1 · 1! + 1!

Problem 22:

1000 · 1000! + 999 · 999! + · · · + 2 · 2! + 1 · 1! = ?

Main Idea: (n + 1)! = n · n! + n!

1001! = 1000 · 1000! + 1000!

= 1000 · 1000! + 999 · 999! + 999!

= 1000 · 1000! + 999 · 999! + 998 · 998! + 998!

= · · · = 1000 · 1000! + · · · + 2 · 2! + 1 · 1!︸︷︷︸?

Problem 22:

1000 · 1000! + · · · + 2 · 2! + 1 · 1! = 1001! − 1

Problem 22:

1000 · 1000! + · · · + 2 · 2! + 1 · 1! = 1001! − 1

= 2002 · k − 1

Problem 22:

1000 · 1000! + · · · + 2 · 2! + 1 · 1! = 1001! − 1

= 2002 · k − 1

= 2002(k − 1) + 2001

Problem 22:

1000 · 1000! + · · · + 2 · 2! + 1 · 1! = 1001! − 1

= 2002 · k − 1

= 2002(k − 1) + 2001

Problem 22:

1000 · 1000! + · · · + 2 · 2! + 1 · 1! = 1001! − 1

= 2002 · k − 1

= 2002(k − 1) + 2001↑

answer

Problem 23:

j = John’s age when Tammy was 4

m = Martha’s age when Tammy was 4

Problem 23:

j = 3m,

Problem 23:

j = 3m, m + a = 2(4 + a),

Problem 23:

j = 3m, m + a = 2(4 + a),

j + a = 5(4 + a),

Problem 23:

j = 3m, m + a = 2(4 + a),

j + a = 5(4 + a), j + b = 2(m + b)

Problem 23:

j = 3m, m + a = 2(4 + a),

j + a = 5(4 + a), j + b = 2(m + b)

4 + b = ?

Problem 23:

j = 3m, m + a = 2(4 + a),

j + a = 5(4 + a), j + b = 2(m + b)

4 + b = ?

Problem 23:

j = 3m, m + a = 2(4 + a),

j + a = 5(4 + a), j + b = 2(m + b)

4 + b = ?

Problem 23:

j = 3m, m + a = 2(4 + a),

j + a = 5(4 + a), j + b = 2(m + b)

4 + b = ?

b = j − 2m =

Problem 23:

j = 3m, m + a = 2(4 + a),

j + a = 5(4 + a), j + b = 2(m + b)

4 + b = ?

b = j − 2m = 3m − 2m = m

Problem 23:

j = 3m, m + a = 2(4 + a),

j + a = 5(4 + a), j + b = 2(m + b)

4 + b = ?

b = j − 2m = 3m − 2m = m

Problem 23:

j = 3m, m + a = 2(4 + a),

j + a = 5(4 + a), j + b = 2(m + b)

4 + b = ?

b = j − 2m = 3m − 2m = m

m = 8 + a,

Problem 23:

j = 3m, m + a = 2(4 + a),

j + a = 5(4 + a), j + b = 2(m + b)

4 + b = ?

b = j − 2m = 3m − 2m = m

m = 8 + a, 3m =

Problem 23:

j = 3m, m + a = 2(4 + a),

j + a = 5(4 + a), j + b = 2(m + b)

4 + b = ?

b = j − 2m = 3m − 2m = m

m = 8 + a, 3m = 20 + 4a

Problem 23:

j = 3m, m + a = 2(4 + a),

j + a = 5(4 + a), j + b = 2(m + b)

4 + b = ?

b = j − 2m = 3m − 2m = m

m = 8 + a, 3m = 20 + 4a

Problem 23:

j = 3m, m + a = 2(4 + a),

j + a = 5(4 + a), j + b = 2(m + b)

4 + b = ?

b = j − 2m = 3m − 2m = m

m = 8 + a, 3m = 20 + 4a

b = m = 4m − 3m =

Problem 23:

j = 3m, m + a = 2(4 + a),

j + a = 5(4 + a), j + b = 2(m + b)

4 + b = ?

b = j − 2m = 3m − 2m = m

m = 8 + a, 3m = 20 + 4a

b = m = 4m − 3m = 4(8 + a) − (20 + 4a)

Problem 23:

j = 3m, m + a = 2(4 + a),

j + a = 5(4 + a), j + b = 2(m + b)

4 + b = ?

b = j − 2m = 3m − 2m = m

m = 8 + a, 3m = 20 + 4a

b = m = 4m − 3m = 4(8 + a) − (20 + 4a) = 12

Problem 23:

j = 3m, m + a = 2(4 + a),

j + a = 5(4 + a), j + b = 2(m + b)

4 + b = 16

b = j − 2m = 3m − 2m = m

m = 8 + a, 3m = 20 + 4a

b = m = 4m − 3m = 4(8 + a) − (20 + 4a) = 12

Remarks on Writing a High School Competition:

Original Problem 23: When Tammy was four years old,John was three times as old as Martha. When Marthawas twice as old as Tammy, John was five times as oldas Tammy. How old was Tammy when John was twice asold as Martha?

Concern: What does it mean to say someone is some age?

Concern: What does it mean to say someone is some age?When someone is 8.75 years old, don’t we say they are 8?

Concern: What does it mean to say someone is some age?When someone is 8.75 years old, don’t we say they are 8?Does that really make a difference in the problem?

Original Problem 23: When Tammy was four years old,John was three times as old as Martha. When Marthawas twice as old as Tammy, John was five times as oldas Tammy. How old was Tammy when John was twice asold as Martha? (Recall Answer: 16)

Tammy John Martha

4 24 8 (really8.75)

Years Later Tammy John Martha

4 24 8 (really8.75)

4 24 8 (really8.75)1.5 5 25 10

4 24 8 (really8.75)1.5 5 25 106.5 12 32 16

Revised Problem (Draft 5): Two days ago, Bobby had4 marbles and Greg had three times as many marbles asPeter. Yesterday, their mother gave each of them some ad-ditional marbles (each boy received the same amount), sothat Peter had twice as many marbles as Bobby, and Greghad five times as many marbles as Bobby. Today, their fa-ther gave each of them some additional marbles (each boyreceived the same amount), so that Greg now has twice asmany marbles as Peter. How many marbles does Bobbyhave today?

Yet Another Version: Let f1, f2, andf3 be the functionsx + a, x + b, andx + c in some order wherea, b, andc

are real numbers. Suppose there are real numbersx1, x2,andx3 satisfying:

(i) f1(x1) = 4 andf3(x1) = 3f2(x1)

(ii) f2(x2) = 2f1(x2) andf3(x2) = 5f1(x2)

(iii) f3(x3) = 2f2(x3)

What is the value off1(x3)?

The Best Idea (my opinion):

Use a different problem!!

Actual Problem 23: Tammy, John, and Martha were allborn at noon on January 19th, but in different years. WhenTammy was four years old, John was three times as old asMartha. When Martha was twice as old as Tammy, Johnwas five times as old as Tammy. How old was Tammywhen John was twice as old as Martha?

Actual Problem 23: Tammy, John, and Martha were allborn at noon on January 19th, but in different years.WhenTammy was four years old, John was three times as old asMartha. When Martha was twice as old as Tammy, Johnwas five times as old as Tammy. How old was Tammywhen John was twice as old as Martha?

Problem 27:

5n + 26

2n + 3is an integer

Problem 27:

5n + 26

2n + 3is an integer

For how many integersn is this the case?

Problem 27:

5n + 26

2n + 3is an integer

2n + 3 divides 5n + 66+ 2

Problem 27:

5n + 26

2n + 3is an integer

2n + 3 divides 5n + 6

⇓2n + 3 divides 2(5n + 26) − 5(2n + 3) = 37

Problem 27:

5n + 26

2n + 3is an integer

m2n + 3 divides 2(5n + 26) − 5(2n + 3) = 37

Problem 27:

5n + 26

2n + 3is an integer

m2n + 3 divides 2(5n + 26) − 5(2n + 3) = 37

2n+3 ∈ {±1, ±37}

Problem 27:

5n + 26

2n + 3is an integer

m2n + 3 divides 2(5n + 26) − 5(2n + 3) = 37

2n+3 ∈ {±1, ±37} and n ∈ {−20, −2, −1, 17}

Problem 27:

5n + 26

2n + 3is an integer

For how many integersn is this the case?4

m2n + 3 divides 2(5n + 26) − 5(2n + 3) = 37

2n+3 ∈ {±1, ±37} and n ∈ {−20, −2, −1, 17}

people.math.sc.edu · Remarks on Writing a High School Competition: Original Problem 23: When Tammy...

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