129 Contest 2014 Solutions

THE UNIVERSITY OF VERMONT

DEPARTMENT OF MATHEMATICS AND STATISTICS FIFTY-SEVENTH ANNUAL HIGH SCHOOL PRIZE EXAMINATION

MARCH 19, 2014 1) Express 1

1 20

1 ₁₄1 as a rational number in lowest terms.

1

1 20 1 ₁₄1

140 140

140 7 140 10

147 150

49

50

2) Express 84 32 3 9 1 2 as a rational number in lowest terms.

84 3_2 3 ₉ 1 2_{ 16}1 8

1 3 16

3 8 24 

16 5 24

20 6

10 3

3) Express log₂125 log₃ 49 log₅81 log₇ 64 as an integer.

log₂125 log₃49 log₅ 81 log₇ 64 3 log₂5 2 log₃ 7 4 log₅ 3 6 log₇2

log₂125 log₃49 log₅ 81 log₇ 64 3 log₂5 2_log2log27₃ 4log2_log23₅ 6_log21₇

log₂125 log₃49 log₅ 81 log₇ 64 3 2 4 6 144

4) The formula for limeade calls for 4 ounces of lime juice for every 12 ounces of water. Karla initially uses 18 ounces of water to make her limeade. If her limeade has 40% too much lime juice, how many ounces of water does she need to add to her mixture to have the correct ratio of lime juice to water? Express the answer as a rational number in lowest terms.

Original lime 18₁₂ 4 1.4 3₂ 4 14₅ 42₅

18 x 342₅

x 342₅ 18 126₅ 90₅ 36 5

5) During the winter, 60% of Vermonters ski and during the summer, 45% of Vermonters hike. If 15% of Vermonters do both activities, what percent of Vermonters do neither?

_{45 15 30}

10

Ski Hike

6 Find the total area of the shaded regions if the area of rectangle

ABCD is 40 square units.

A B

C D

1 2 x

1 2 y

y

x

21₂ 1₂x1₂y 1₄x y 1₄ 40 10

7) A candy store sold bags of 40 caramels for $3.20, bags of 40 chocolates for $4.00 and mixed bags of chocolates and caramels for $3.50. If the mixed bags also have 40 pieces of candy, how many caramels are in each mixed bag?

x3.20₄₀  40 x 4.00₄₀ 350

320x 40 x 400 350 40 320x 40 400 400x 40(350) 80x 40 350 400 40 50

x 40 50₈₀ 25

8) The function f satisfies 2f x 6f1

x x

2 _{for all x 0. Find f(2) and express the}

answer as a rational number in lowest terms.

2f 2 6f

1 2 4

2f1₂ 6f 2 1₄

f 2 3f

1 2 2

f1₂ 3f 2 1₈

f 2 3f

1 2 2

3f1₂ 9f 2 3₈

8f 2 19₈

f 2 –19 64

9) Find the real number x such that log₃ x 4 log_x 9 4.

1₂log₃x 8 logx 3 4

1₂log₃x _{8 log3}1 _x 4

y 16

y 8

y2 8y 16 0

y 4 2 _{0 y 4 log}

3x 4 x 81

10) Suppose that f is a function such that f 3x ₃3

x for all real x 0.

Determine the value of f 10 .

f 10 f310₃  3

3 10₃ 9 9 10

9 19

11) Express 12510 2510

1254 ₂₅11 in simplest form.

12510 2510

1254 ₂₅11

530 ₅20 512 ₅22

520_510 _1 512_510 _1 5

8 ₅4 ₆₂₅

12) In how many ways can 24 cents be paid using any combination of pennies, nickels and dimes?

10 5 1

20 0 9

10 0 14

10 5 9

10 10 4

0 0 24

0 5 19

0 10 14

0 15 9

0 20 4

n 9

13) Find all real values of x such that x 2 3 4.

| | x – 2 | – 3 | = 4 | x – 2 | – 3 = 4 or | x – 2 | – 3 = – 4

| x – 2 | = 7 | x – 2 | = – 1 Impossible x – 2 = 7 or x – 2 = – 7

x = 9 or x – 5

13) If the number 15! written in base 12 ends in k zeros, what is the value of k?

14) The average of a set of 50 numbers is 45 and the average of a set of mnumbers is 65. If the average of the combined sets is 60, what is m?

50 45_{50 m}m65 60

50 45 65m 50 60 60m 5m 50 15

m 3 150 150

15) Express sin2 19 cos2 26 1₂sin 38 sin 52 sin2 26 cos2 19 as a rational

number in lowest terms.

sin2 19 cos226 1₂sin 38 sin 52 sin226 cos2 19

sin2_{19 cos}2 ₂₆ 1

2sin 2 19 sin 2 26 sin

2_{26 cos}2 ₁₉

sin219 cos226 1₂2 sin 19 cos 19 2 sin 26 cos 26 sin226 cos219

sin2_{19 cos}2_{26 2 sin 19 cos 19 sin 26 cos 26 sin}2_{26 cos}2 ₁₉

sin 19 cos 26 sin 26 cos 19 2

sin 45 2 1 2

2

1 2

16) The probability that Sheila hits the bullseye when playing darts is 1₄. If she tosses

three darts, what is the probability she will hit the bullseye at least once? Express your answer as a rational number in lowest terms.

Hits 1 31

4 3 4

2 ₂₇ 64

Hits 2 31

4 2

3 4

9 64

Hits 3 1

4 3

1 64

37 64

p 37

64

17) Find the coordinates of the center of the circle that passes through the points

7, 0 , 2, 1 and 2, 5 . Express the answer as an ordered pair of real numbers a,b.

7 h 2 0 k2 r2 2 h2 1 k2 r2 2 h2 5 k2 r2

1 49 14h h2 k2 r2

2 4 4h h2 1 2k k2 r2 3 4 4h h2 25 10k k2 r2

2 1 45 10h 1 2k 0

3 2 24 8k 0 k 3 h 5

h,k 5, 3

18) If x and y satisfy 1_x 1_y 1₂ and xy 6, find the value of x3 y3.

1_x 1_y 1₂ x_{x y}y 1₂ x y 1₂ xy 1₂ 6 3

x_xyy3 1₈

x3 3x2y 3xy2 y3

xy3

1 8

x3 3x2y 3xy2 y3 1₈ 6 2

x3 y3 3xy x y 1₈ 63

19) Ticket prices for a local community orchestra are $15 for adults, $12 for seniors and $7 for students. At a recent concert, the orchestra sold 120 tickets for a total of $1481. What is the maximum possible number of student tickets that were sold?

Let A, SandT be the numbers of adult, senior and student tickets sold.

 A S T 120 15A 12S 7T 1481

Solving for AandS A 41 5₃ T andS 319 8₃ T

41 5T 3Αand 319 8T 3Β for non negative integersΑandΒ 319 8T 0 T 39

T

41 5T 319–8T

39 236 7

38 231 15

Thus T 38

20) Let a0 2,a1 5 and an an 1 an 2 for n 2. Find n 0

2014

an .

a0 2

a1 5

a2 3

a3 2

a4 1

a5 1

a6 0

a7 1

a8 1

a9 0

Thus, starting with a4 each 3 consecutive terms asum to 2.

n 0 2014

an a0 a1 a2 a3 n 4

2014

an a0 a1 a2 a3 n 4

2013

an a2014

n 0 2014

an 2 5 3 2 670 2 1 1353

21) What is the minimum value of 9 25t6

t3 , where t is a positive real number ?

9 25t6

t3

9 5t3_2 _30t3 t3

9 5t3_2

t3 30

Thus the minimum value occurs when 9 5t3 0. Minimum = 30 22) Let R be the region in the x y plane bounded by the line segments joining 0, 0 , 0, 5 , 4, 5 , 4, 1 , 7, 1 , 7, 0 and 0, 0 , in the given order.

The line y k x divides R into two subregions of equal area. Determine the value of k.

0, 0 7, 0

7, 1 4, 1

4, 5 0, 5

2 4 6

1 2 3 4 5

Total area = 5 4 3 1 23

From the upper trapezoid 4 5 5 4₂ k 23₂

4 10 4k 23 40 16k 23 16k 17 k 17

16

23) When a complex number z is expressed in the form z a b i, wherei2 1 and aandbare real numbers, the modulus of z, denoted z , is defined by

z a2 b2 . Find all complex numbers z of modulus 1 such that 3 4i z is a real number. Express your answer(s) in the form c d i, wherec and d are real numbers.

3 4i a bi 3a 4b 4a 3b i

For the product to be real 4a 3b 0 a 3₄b

z 1 a2 b2 1  3₄b2 b2 1 ₁₆9 1b2 1

25₁₆b2 1 b2 16₂₅ b 4₅

a bi 3

5 4

5i or 3

5 4

5i

24 In ABC, AB AC and point Q strictly between A and B

is located on AB so that AQ QC CB. Determine the degree

measure of angle A.

A

B C

Q

Α

Α Β

Β Β Α

AQ QC QAC QCA Α

AB AC ABC ACB Β QCB Β Α Β an exterior angle of QAC Β 2Α

From Q C B Β Β Β Α 180 3Β Α 180 3 2Α a 180 5Α 180

Α 36

25) For a real number x, define f x 16x x2 30x x2 224 . Determine the largest possible positive value of f x.

16x x2 30x x2 224 x16 x x 14 x 16

Both radicands are positive for 14 x 16. Max occurs when the first is largest and the second smallest. i.e. x 14

f 14 14 16 14 14 2 2 7

26) In a list of the base 4 representations of the decimal integers from 0 to 1023, the digit 3 appears a total of k times. Find k.

The decimal integers 0 to 1023 in base 4 can be represented as 00 0004 33 3334

Considering each of the base 4 integers as 5 digits, the total number of digits is 5(1024).

Since each digit appears an equal number of times, the number of 3s is 5 1024₄ 1280

27 Circles CPand CQwith centers at P and Q are externally

tangent and have radii 2 and 1, respectively. Line segment

AC is tangent to circle CPat A and line segment BC is tangent

to CPand CQat T and S, respectively. Find the length AC.

A B

C

S T

P Q

Β x 1 1 2 2

From Q S B sinΒ ₁1_x

From P T B sinΒ ₄2

x

₁1_{x 4}2_x 4 x 2 2x x 2

From Q S B sinΒ _{1 2}1 1₃ tanΒ 1

8

From AB C tan Β ₆AC_x AC₈ 1

8

AC 8

8

8 2 2

28 Suppose that A and B are points on a circle with center O . If the perimeter of sector OAB is 10 units and the area of sector OAB is 4 square units, find all

possible values of the length of arc AB. _O

A B

Θ r

s

arcAB s rΘ

10 2r rΘ r 10₂rΘ 5 r₂Θ

4 1₂r2Θ r rΘ 8

5 r₂Θ rΘ 8 10rΘ r₂Θ2 16 rΘ 8 rΘ 2 0 rΘ 2, 8

29) If the roots of x2 ax b 0 are the cubes of the roots of x2 x 2 0, find aandb. Let r1 andr2 be the roots of x2 x 2 0 r1r2 2 and r1 r2 1

r13 and r23 roots of x2 ax b 0 r13r23 b and r13 r23 a

r13r23 r1r2 3 23 8

r1 r23 r13 3r12r2 3r1r23 r23 1

r13 r23 3r1r2r1 r2 1 r13 r23 1 3 2 1 5

a,b 5, 8

30) How many positive integers x have the property that 14 is the remainder when 2014 is divided by x ?

2014 qx 14 qx 2000 soxdivides 2000 andx 14.

2000 24 53 4 1 3 1 20 divisors. Of these 1, 2, 4, 8, 5, 10 are less than 14.

The number of x satisfying the given conditions is 20 6 14

31) Find the smallest positive value of x (in radians) such that tan 2x cos_cosx sinx

x sinx .

tan 2x cos_cosx sinx

x sinx

_{cos 2}sin 2x_{x cos}cosx_x sin_sinx_x cos_cosxx _sinsinx_x cos2x 2 sinx cosx sin2x

cos2_{x sin}2_x

_{cos 2}sin 2x_x 1 sin 2_{cos 2x}x

sin 2x 1 sin 2x 2 sin 2x 1 sin 2x 1₂ 2x Π₆ x Π 12

32 How many paths are there from A to B, if at each intersection you can only move in the indicated direction s ?

B

A

1 2 6 22 22

1 4 16 60

1 6 82

1 89

90

Number of paths = 90

33) If x2 6x 6 0 , what is the value of x3 7x2 2014? x2 6x 6 0 x2 6x 6

x3 7x2 2014 xx2 6x x 2014 x3 7x2 2014 x6 x 2014 x3 7x2 2014 6 2014 2020 34) For how many integers n is 20 n

14 n an integer?

20 n

14 n 1

6

14 n

14 n divides 6 14 n 1, 2, 3, 6

Thus 8 values of n.

35) The integer M consists of 500 threes and the integer N consists of 500 sixes. What is the sum of the digits in the base 10 representation of the product M N ?

M N 10500₃ 1 210500₃ 1 2₉10500 _12

M N 2₉101000 _{1 210}500 _1

M N 2101000₉ 1 210500₉ 1

M N 2 111 1 2 111 1

M N 222 2 – 444 4 1000 2s minus 500 4s

222 222 2222

– 444 4444

222 217 7778

From left to right: 499 twos, 1 one, 499 sevens and 1 eight.

499 2 1 499 7 8 499 2 7 9 499 9 9 500 9 4500

36) What is the smallest integer that is greater than 5 3 6 ?

 6

 6

 5

   4

 2

 3

 3

 2

 4

   5

 5 3 6

 5

6

6 5  5

 3 15 5

4

 3

2

20 5 3

 3

3

15 5 2

 3

4

6 5  3 5

 3

6

 5 3

6

 5

6

6 5  5

 3 15 5

4

 3

2

20 5 3

 3

3

15 5 2

 3

4

6 5  3 5

 3

6

 5 3

6

 5 3

6

2 5 6

0 15 5 4

 3 

2

0 15 5  2

 3

4

0  3 6

5 3

6

5 3

6

2 125 12 25 3 15 5 9 27

5 3

6

5 3

6

2 125 1125 675 27 2 1952 3904

Since 5 3

6

1 5 3

6

3904

37) The geometric mean of a set of k positive real numbers x1,x2,x3, ,xk is

x1 x2 xk 1k. Find the positive integer n such that the geometric mean of the

set of all positive integer divisors of n is 70.

If n has prime factorization n p1Α1 p2Α2 pjΑj, the number of divisors of n is

k Α1 1 Α2 1 Αj 1 .

If all of the Αi are even, k is odd and k₂1 of the divisors of n appear in pairs xiand _xn₁ with the remaining

divisor n. Then the product x1 x2 xk nk 1 2 n1 2 nk2 and x1 x2 xk 1k n1 2.

If not all of the Αi are even, k is even and the divisors of n appear in pairs xiand_xn 1

Then the product x1 x2 xk nk2 and x1 x2 xk 1k n1 2.

Thus n1 2 70 n 4900

38 An equilateral triangle with side length one is divided into four congruent triangles and the central triangle is shaded. Let the shaded area be

A1. The remaining three triangles are similarly

divided and each central triangle is shaded; the area of the three shaded triangles is A2. This process is

continued . The shaded areas A1, A2and A3are

shown. Find

n 1

An

The area of an equilateral triangle of side length S is A s₄2 3 . Assuming that the original triangle has side lenght S,

The first shaded triangle has side length S₂ and A1 S₂2 ₄3

The nine shaded triangles in A3 have side length S₈ and A3 S₈2 3 2 ₃

4

Continuing this pattern:

!

n 1

"

An S₂2 ₄3 S₄2 3 ₄3 S₈2 3 2 ₃

4

!

n 1

"

An S 2 42 3 1

3

22 32

24  Using the sum of the geormetric series with r 3 4

!

n 1

"

An S 2

42 3

1 1 3

4 s2 ₃

4

With s 1

n 1

An

3

4

39) How many different rectangles can be formed using edges in the left-hand figure below? Two such rectangles are shown in the right-hand figure.

27 21 15 12 6 0 22 17 12 10 5 0 17 13 9 8 4 0 12 3 0 6 3 0 10 8 6 4 2 0 5 4 3 2 1 0 0 0 0 0 0 0

75 66 51 24 30 15

Labeling each node with the number of rectangles with the node as lower left corner and adding:

40 A grassy park in the shape of an equilateral triangle is to be surrounded by a gravel

walkway whose outside edges form an equilateral triangle. If the parallel sides of the walkway are 2 meters apart and the area of the grassy park

is 30,000 3 square meters, what is the area

of the gravel walkway ?

2 2

2

2 2

2

S2 S1

30 2

4

2 3

Let the sides of the inside and outside triantles be S1andS2 respectively.

The areas are respectively ₄3 S1 2and ₄3 S2 2. Hence the area of the walkway is

Aw ₄3 S2 2 ₄3 S12.

From the diagram above we can see that S2 S1 4 3 .

Aw

S1 4 3 2

4 3

S12

4 3

3 4 S1

2 _{8 3 S}

1 48 S12

Aw 3 2 3 S1 12

Using the given area of the inside triangle: ₄3 S1 2 30, 000 3 S1 200 3

41 In a square of side length 2 , each side is divided into 10 equal pieces by inserting 9 equally spaced points. The

corresponding points on adjacent sides are connected by straight line segments as indicated in the figure. Find the sum of the lengths of these diagonal line segments.

s

10

s

10

s

10

L1 L2 L3

L1 = ₁₀s 2 ₁₀s 2 = ₁₀s 2

L2 = 2₁₀s2 ₁₀2s2 = 2₁₀s 2

Lk =  k s 10

2

k s 10

2

= k s₁₀ 2 k = 1, 2, 3, 10

T = 2

k 1 9

Lk + L10 = 2 2 ₁₀s 9 10₂ 1 = 10 2 s