2005 Purple Comet Problems

Part of Purple Comet Problems

Subcontests

(25)

1

Problem 25, Spring 2005 , HS

Find the number of quadruples

(a,b,c,d)

of integers which satisfy both \begin{align*}\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} &= \frac{1}{2}\qquad\text{and}\\\\2(a+b+c+d) &= ab + cd + (a+b)(c+d) + 1.\end{align*}

1

Problem 24, Spring 2005 , HS

\triangle ABC

240

X, Y, Z

AB

BC

CA

, respectively. Given that

\frac{AX}{BX} = 3

\frac{BY}{CY} = 4

\frac{CZ}{AZ} = 5

, find the area of

\triangle XYZ

.
[asy] size(175); defaultpen(linewidth(0.8)); pair A=(0,15),B=(0,-5),C=(25,0.5),X=origin,Y=(4C+B)/5,Z=(5A+C)/6; draw(A--B--C--cycle^^X--Y--Z--cycle); label("

A

",A,N); label("

B

",B,S); label("

C

",C,E); label("

X

",X,W); label("

Y

",Y,S); label("

Z

1

Problem 23, Spring 2005 , HS

a = \sqrt[401]{4} - 1

n \ge 2

b_n = \binom{n}{1} + \binom{n}{2} a + \ldots + \binom{n}{n} a^{n-1}

b_{2006} - b_{2005}

1

Problem 22, Spring 2005 , HS

d_k

be the greatest odd divisor of

k

k = 1, 2, 3, \ldots

d_1 + d_2 + d_3 + \ldots + d_{1024}

1

Problem 21, Spring 2005 , HS

In the diagram below

\angle CAB, \angle CBD

\angle CDE

are all right angles with side lengths

AC = 3

BC = 5

BD = 12

DE = 84

. The distance from point

E

AB

can be expressed as the ratio of two relatively prime positive integers,

m

n

m + n

.
[asy] size(300); defaultpen(linewidth(0.8)); draw(origin--(3,0)--(0,4)--cycle^^(0,4)--(6,8)--(3,0)--(30,-4)--(6,8)); label("

A

",origin,SW); label("

B

",(0,4),dir(160)); label("

C

",(3,0),S); label("

D

",(6,8),dir(80)); label("

E

",(30,-4),E);[/asy]

1

Problem 20, Spring 2005 , HS

\sum_{k=1}^{360} \frac{1}{k \sqrt{k+1} + (k+1)\sqrt{k}}

is the ratio of two relatively prime positive integers

m

n

m + n

1

Problem 19, Spring 2005 , HS

x

y

be integers satisfying both

x^2 - 16x + 3y = 20

y^2 + 4y - x = -12

x + y

1

Problem 18, Spring 2005 , HS

The side lengths of a trapezoid are

\sqrt[4]{3}, \sqrt[4]{3}, \sqrt[4]{3}

2 \cdot \sqrt[4]{3}

. Its area is the ratio of two relatively prime positive integers,

m

n

m + n

1

Problem 17, Spring 2005 , HS

f

g

are defined so that

f(1) = 4

g(1) = 9

, and for each integer

n \ge 1

f(n+1) = 2f(n) + 3g(n) + 2n

g(n+1) = 2g(n) + 3 f(n) + 5

f(2005) - g(2005)

1

Problem 16, Spring 2005 , HS

A tailor met a tortoise sitting under a tree. When the tortoise was the tailor’s age, the tailor was only a quarter of his current age. When the tree was the tortoise’s age, the tortoise was only a seventh of its current age. If the sum of their ages is now

264

, how old is the tortoise?

1

Problem 3, Spring 2005 , HS

Four rectangular strips each measuring

4

16

inches are laid out with two vertical strips crossing two horizontal strips forming a single polygon which looks like a tic-tack-toe pattern. What is the perimeter of this polygon?
[asy] size(100); draw((1,0)--(2,0)--(2,1)--(3,1)--(3,0)--(4,0)--(4,1)--(5,1)--(5,2)--(4,2)--(4,3)--(5,3)--(5,4)--(4,4)--(4,5)--(3,5)--(3,4)--(2,4)--(2,5)--(1,5)--(1,4)--(0,4)--(0,3)--(1,3)--(1,2)--(0,2)--(0,1)--(1,1)--(1,0)); draw((2,2)--(2,3)--(3,3)--(3,2)--cycle); [/asy]

2

Problem 15, Spring 2005 , MS

And it came to pass that Jeb owned over a thousand chickens. So Jeb counted his chickens. And Jeb reported the count to Hannah. And Hannah reported the count to Joshua. And Joshua reported the count to Caleb. And Caleb reported the count to Rachel. But as fate would have it, Jeb had over-counted his chickens by nine chickens. Then Hannah interchanged the last two digits of the count before reporting it to Joshua. And Joshua interchanged the first and the third digits of the number reported to him before reporting it to Caleb. Then Caleb doubled the number reported to him before reporting it to Rachel. Now it so happens that the count reported to Rachel was the correct number of chickens that Jeb owned. How many chickens was that?

Problem 15, Spring 2005 , HS

And it came to pass that Jeb owned over a thousand chickens. So Jeb counted his chickens. And Jeb reported the count to Hannah. And Hannah reported the count to Joshua. And Joshua reported the count to Caleb. And Caleb reported the count to Rachel. But as fate would have it, Jeb had over-counted his chickens by nine chickens. Then Hannah interchanged the last two digits of the count before reporting it to Joshua. And Joshua interchanged the first and the third digits of the number reported to him before reporting it to Caleb. Then Caleb doubled the number reported to him before reporting it to Rachel. Now it so happens that the count reported to Rachel was the correct number of chickens that Jeb owned. How many chickens was that?

2

Problem 14, Spring 2005 , MS

Eight identical cubes with of size

1 \times 1 \times 1

each have the numbers

1

6

written on their faces with the number

1

written on the face opposite number

2

3

written on the face opposite number

5

4

written on the face opposite number

6

. The eight cubes are stacked into a single

2 \times 2 \times 2

cube. Add all of the numbers appearing on the outer surface of the new cube. Let

M

be the maximum possible value for this sum, and

N

be the minimum possible value for this sum. Find

M - N

Problem 14, Spring 2005 , HS

Four mathletes and two coaches sit at a circular table. How many distinct arrangements are there of these six people if the two coaches sit opposite each other?

2

Problem 13, Spring 2005 , MS

x

\frac{\frac{5}{x-50}+ \frac{7}{x+25}}{\frac{2}{x-50}- \frac{3}{x+25}} = 17.

Problem 13, Spring 2005 , HS

The work team was working at a rate fast enough to process

1250

items in ten hours. But after working for six hours, the team was given an additional

165

items to process. By what percent does the team need to increase its rate so that it can still complete its work within the ten hours?

2

Problem 11, Spring 2005 , MS

The work team was working at a rate fast enough to process

1250

items in ten hours. But after working for six hours, the team was given an additional

150

items to process. By what percent does the team need to increase its rate so that it can still complete its work within the ten hours?

Problem 11, Spring 2005 , HS

The straight river is one and a half kilometers wide and has a current of

8

kilometers per hour. A boat capable of traveling

10

kilometers per hour in still water, sets out across the water. How many minutes will it take the boat to reach a point directly across from where it started?

2

Problem 12, Spring 2005 , HS

Find the number of nonnegative integers

n

(n^2 - 3n +1)^2 + 1

is a prime number.

Problem 12, Spring 2005 , MS

Four mathletes and two coaches sit at a circular table. How many distinct arrangements are there of these six people if the two coaches sit opposite each other?

2

Problem 10, Spring 2005 , HS

2

yellow candies,

4

red candies, and

6

blue candies. Candies are randomly drawn out of the jar one-by-one and eaten. The probability that the

2

yellow candies will be eaten before any of the red candies are eaten is given by the fraction

\frac{m}{n}

m

n

are relatively prime positive integers. Find

m + n

Problem 10, Spring 2005 , MS

1000

th digit to the right of the decimal point in the decimal representation of

\tfrac{37}{5500}

2

Problem 9, Spring 2005 , MS

Find the number of nonnegative integers

n

(n^2 - 3n + 1)^2 + 1

is a prime number

Problem 9, Spring 2005 , HS

T

30-60-90

triangle with hypotenuse of length

20

. Three circles, each externally tangent to the other two, have centers at the three vertices of

T

. The area of the union of the circles intersected with

T

(m + n \sqrt{3}) \pi

for rational numbers

m

n

m + n

2

Problem 8, Spring 2005 , MS

1

is special. The number

2

is special because it is relatively prime to

1

3

is not special because it is not relatively prime to the sum of the special numbers less than it,

1 + 2

4

is special because it is relatively prime to the sum of the special numbers less than it. So, a number bigger than

1

is special only if it is relatively prime to the sum of the special numbers less than it. Find the twentieth special number.

Problem 8, Spring 2005 , HS

x

\cfrac{1}{\cfrac{1}{\cfrac{1}{\cfrac{1}{x}+\cfrac12}+\cfrac{1}{\cfrac{1}{x}+\cfrac12}}+\cfrac{1}{\cfrac{1}{\cfrac{1}{x}+\cfrac12}+\cfrac{1}{\cfrac{1}{x}+\cfrac12}}}=\frac{x}{36}.

2

Problem 7, Spring 2005 , MS

Bill’s age is one third larger than Tracy’s age. In

30

years Bill’s age will be one eighth larger than Tracy’s age. How many years old is Bill?

Problem 7, Spring 2005 , HS

The graph of the equation

y = 5x + 24

intersects the graph of the equation

y = x^2

at two points. The two points are a distance

\sqrt{N}

N

2

Problem 6, Spring 2005 , MS

We glue together

990

one inch cubes into a

9

10

11

inch rectangular solid. Then we paint the outside of the solid. How many of the original

990

cubes have just one of their sides painted?

Problem 6, Spring 2005 , HS

ABCDE

is a regular pentagon. What is the degree measure of the acute angle at the intersection of line segments

AC

BD

2

Problem 5, Spring 2005 , MS

A palindrome is a number that reads the same forwards and backwards such as

3773

42924

. Find the sum of the twelve smallest five digit palindromes.

Problem 5, Spring 2005 , HS

In January Jeff’s investment went up by three quarters. In February it went down by one quarter. In March it went up by one third. In April it went down by one fifth. In May it went up by one seventh. In June Jeff’s investment fell by

\frac{m}{n}

m

n

are relatively prime positive integers. If Jeff’s investment was worth the same amount at the end of June as it had been at the beginning of January, find

m + n

2

Problem 4, Spring 2005 , MS

Fill in numbers in the boxes below so that the sum of the entries in each three consecutive boxes is

2005

. What is the number that goes into the leftmost box?
[asy] size(300); label("999",(2.5,.5)); label("888",(7.5,.5));
draw((0,0)--(9,0)); draw((0,1)--(9,1)); for (int i=0; i<=9; ++i) { draw((i,0)--(i,1)); } [/asy]

Problem 4, Spring 2005 , HS

A palindrome is a number that reads the same forwards and backwards such as

3773

42924

. What is the smallest

9

digit palindrome which is a multiple of

3

and has at least two digits which are

5

's and two digits which are

7

2

Problem 2, Spring 2005 , MS

Jerry is mowing a rectangular lawn which is

77

feet north to south by

83

feet east to west. His lawn mower cuts a path

18

inches wide. Jerry mows the grass by cutting a path from west to east across the north side of the lawn and then making a right turn cutting a path along the east side of the lawn. When he completes mowing each side of the lawn, he continues by making right turns to mow a path along the next side. How many right turns will he make?

Problem 2, Spring 2005 , HS

We glue together

990

one inch cubes into a

9

10

11

inch rectangular solid. Then we paint the outside of the solid. How many of the original

990

cubes have just one of their sides painted?

2

Problem 1, Spring 2005 , MS

A cubic inch of the newly discovered material madelbromium weighs

5

ounces. How many pounds will a cubic yard of madelbromium weigh?

Problem 1, Spring 2005 , HS

The cost of producing each item is inversely proportional to the square root of the number of items produced. The cost of producing ten items is

\$2100

. If items sell for

\$30

each, how many items need to be sold so that the producers break even?