2017 Purple Comet Problems

Part of Purple Comet Problems

Subcontests

(30)

1

Purple Comet 2017 HS problem 30

A container is shaped like a right circular cone with base diameter

18

12

. The vertex of the container is pointing down, and the container is open at the top. Four spheres, each with radius

3

, are placed inside the container as shown. The first sphere sits at the bottom and is tangent to the cone along a circle. The second, third, and fourth spheres are placed so they are each tangent to the cone and tangent to the rst sphere, and the second and fourth spheres are each tangent to the third sphere. The volume of the tetrahedron whose vertices are at the centers of the spheres is

K

K^2

. https://cdn.artofproblemsolving.com/attachments/9/c/648ec2cf0f0c2f023cd00b1c0595a9396d0ddc.png

1

Purple Comet 2017 HS problem 21

The diagram below shows a large circle. Six congruent medium-sized circles are each internally tangent to the large circle and tangent to two neighboring medium-sized circles. Three congruent small circles are mutually tangent to one another and are each tangent to two medium-sized circles as shown. The ratio of the area of the large circle to the area of one of the small circles can be written as

m+\sqrt{n}

m

n

are positive integers. Find

m + n

. https://cdn.artofproblemsolving.com/attachments/a/4/fcffd7ee6e8d3da0641525e7a987d13ce05496.png

1

Purple Comet 2017 HS problem 26

The incircle of

\vartriangle ABC

is tangent to sides

\overline{BC}, \overline{AC}

\overline{AB}

D, E

F

, respectively. Point

G

is the intersection of lines

AC

DF

as shown. The sides of

\vartriangle ABC

AB = 73, BC = 123

AC = 120

. Find the length

EG

. https://cdn.artofproblemsolving.com/attachments/d/a/aede28071a1a6b94bbe3ad8e1e104822b89439.png

1

Purple Comet 2017 HS problem 29

Find the number of three-element subsets of

\{1, 2, 3,...,13\}

that contain at least one element that is a multiple of

2

, at least one element that is a multiple of

3

, and at least one element that is a multiple of

5

\{2,3, 5\}

\{6, 10,13\}

1

Purple Comet 2017 HS problem 28

T_k = \frac{k(k+1)}{2}

k

-th triangular number. The infinite series

\sum_{k=4}^{\infty}\frac{1}{(T_{k-1} - 1)(Tk - 1)(T_{k+1} - 1)}

\frac{m}{n}

m

n

are relatively prime positive integers. Find

m + n

1

Purple Comet 2017 HS problem 27

Find the minimum value of

4(x^2 + y^2 + z^2 + w^2) + (xy - 7)^2 + (yz - 7)^2 + (zw - 7)^2 + (wx - 7)^2

x, y, z

w

range over all real numbers.

1

Purple Comet 2017 HS problem 25

Leaving his house at noon, Jim walks at a constant rate of

4

miles per hour along a

4

mile square route returning to his house at

1

PM. At a randomly chosen time between noon and

1

PM, Sally chooses a random location along Jim's route and begins running at a constant rate of

7

miles per hour along Jim's route in the same direction that Jim is walking until she completes one

4

mile circuit of the square route. The probability that Sally runs past Jim while he is walking is given by

\frac{m}{n}

m

n

are relatively prime positive integers. Find

m + n

1

Purple Comet 2017 HS problem 24

Eight red boxes and eight blue boxes are randomly placed in four stacks of four boxes each. The probability that exactly one of the stacks consists of two red boxes and two blue boxes is

\frac{m}{n}

, where m and n are relatively prime positive integers. Find

m + n

1

Purple Comet 2017 HS problem 23

3

-dimensional cube has

6

2

-dimensional faces,

12

1

-dimensional edges, and

8

0

-dimensional vertices. Find the number of

9

-dimensional sub-subfaces in a

12

-dimensional cube.

1

Purple Comet 2017 HS problem 22

Find the number of functions

f

that map the set

\{1,2, 3,4\}

into itself such that the range of the function

f(x)

is the same as the range of the function

f(f(x))

2

Purple Comet 2017 MS Problem 20

A right circular cone has a height equal to three times its base radius and has volume 1. The cone is inscribed inside a sphere as shown. The volume of the sphere is

\frac{m}{n}

m

n

are relatively prime positive integers. Find

m + n

.
https://snag.gy/92ikv3.jpg

Purple Comet 2017 HS problem 20

a

be a solution to the equation

\sqrt{x^2 + 2} = \sqrt[3]{x^3 + 45}

. Evaluate the ratio of

\frac{2017}{a^2}

a^2 - 15a + 2

2

Purple Comet 2017 MS Problem 19

Find the sum of all values of

a + b

(a, b)

is an ordered pair of positive integers and

a^2+\sqrt{2017-b^2}

is a perfect square.

Purple Comet 2017 HS problem 19

Find the greatest integer

n < 1000

4n^3 - 3n

is the product of two consecutive odd integers.

2

Purple Comet 2017 MS Problem 18

Omar has four fair standard six-sided dice. Omar invented a game where he rolls his four dice over and over again until the number 1 does not appear on the top face of any of the dice. Omar wins the game if on that roll the top faces of his dice show at least one 2 and at least one 5. The probability that Omar wins the game is

\frac{m}{n}

m

n

are relatively prime positive integers. Find

m + n

Purple Comet 2017 HS problem 18

3

-dimensional coordinate space nd the distance from the point

(36, 36, 36)

to the plane that passes through the points

(336, 36, 36)

(36, 636, 36)

(36, 36, 336)

2

Purple Comet 2017 MS Problem 17

a_0

a_1

a_6

be real numbers such that

a_0 + a_1 + ... + a_6 = 1

a_0 + a_2 + a_3 + a_4 + a_5 + a_6 =\frac{1}{2}

a_0 + a_1 + a_3 + a_4 + a_5 + a_6 = \frac{2}{3}

a_0 + a_1 + a_2 + a_4 + a_5 + a_6 =\frac{7}{8}

a_0 + a_1 + a_2 + a_3 + a_5 + a_6 =\frac{29}{30}

a_0 + a_1 + a_2 + a_3 + a_4 + a_6 =\frac{143}{144}

a_0 + a_1 + a_2 + a_3 + a_4 + a_5 =\frac{839}{840}

a_0

\frac{m}{n}

m

n

are relatively prime positive integers. Find

m + n

Purple Comet 2017 HS problem 17

\left(1 + \sqrt[6]{26 + 15\sqrt3} -\sqrt[6]{26 - 15\sqrt3}\right)^6= m + n\sqrt{p}

m, n

p

are positive integers, and

p

is not divisible by the square of any prime. Find

m + n + p

2

Purple Comet 2017 MS Problem 16

The set of positive real numbers

x

that satisfy 2 | x^2 - 9 | \le 9 | x | is an interval

[m, M]

10m + M

Purple Comet 2017 HS problem 16

a_1 = 1 +\sqrt2

n \ge 1

a_{n+1} = 2 -\frac{1}{a_n}

. Find the greatest integer less than or equal to the product

a_1a_2a_3 ... a_{200}

2

Purple Comet 2017 MS Problem 15

Find the remainder when

7^{7^7}

1000

Purple Comet 2017 HS problem 15

For real numbers

a, b

c

p(x) = 3x^7 - 291x^6 + ax^5 + bx^4 + cx^2 + 134x - 2

7

real roots whose sum is

97

. Find the sum of the reciprocals of those

7

2

Purple Comet 2017 MS Problem 14

Let a and b be positive integers such that

a + ab = 1443

ab + b = 1444

10a + b

Purple Comet 2017 HS problem 14

Find the sum of all integers

n

n - 3

n^2 + 4

are both perfect cubes.

2

Purple Comet 2017 MS Problem 13

Find the number of positive integer divisors of

20^{17}

that are either perfect squares or perfect cubes.

Purple Comet 2017 HS problem 13

ABCDE

be a pentagon with area

2017

such that four of its sides

AB, BC, CD

EA

have integer length. Suppose that

\angle A = \angle B = \angle C = 90^o

AB = BC

CD = EA

. The maximum possible perimeter of

ABCDE

a + b \sqrt{c}

a

b

c

are integers and

c

is not divisible by the square of any prime. Find

a + b + c

2

Purple Comet 2017 MS Problem 12

x

y

z

be real numbers such that

12x - 9y^2 = 7

6y - 9z^2 = -2

12z - 9x^2 = 4

6x^2 + 9y^2 + 12z^2

Purple Comet 2017 HS problem 12

P

be a polynomial satisfying

P(x + 1) + P(x - 1) = x^3

for all real numbers

x

. Find the value of

P(12)

2

Purple Comet 2017 MS Problem 11

Find the greatest prime divisor of

29! + 33!

Purple Comet 2017 HS problem 11

Dave has a pile of fair standard six-sided dice. In round one, Dave selects eight of the dice and rolls them. He calculates the sum of the numbers face up on those dice to get

r_1

. In round two, Dave selects

r_1

dice and rolls them. He calculates the sum of the numbers face up on those dice to get

r_2

. In round three, Dave selects

r_2

dice and rolls them. He calculates the sum of the numbers face up on those dice to get

r_3

. Find the expected value of

r_3

2

Purple Comet 2017 MS Problem 10

Find the number of rearrangements of the letters in the word MATHMEET that begin and end with the same letter such as TAMEMHET.

Purple Comet 2017 HS problem 10

Find the number of positive integers less than or equal to

2017

that have at least one pair of adjacent digits that are both even. For example, count the numbers

24

1862

2012

4

58

1276

2

Purple Comet 2017 MS Problem 9

\triangle{ADE}

B

C

AD

F

G

AE

BG \parallel CF \parallel DE

, as shown. The area of

\triangle{ABG}

36

, the area of trapezoid

CFED

144

AB = CD

. Find the area of trapezoid

BGFC

.
https://snag.gy/SIuOLB.jpg

Purple Comet 2017 HS problem 9

The diagram below shows

\vartriangle ABC

D

\overline{BC}

. Three lines parallel to side

\overline{BC}

\overline{AD}

into four equal segments. In the triangle, the ratio of the area of the shaded region to the area of the unshaded region is

\frac{49}{33}

\frac{BD}{CD} = \frac{m}{n}

m

n

are relatively prime positive integers. Find

m + n

. https://cdn.artofproblemsolving.com/attachments/2/8/77239633b68f073f4193aa75cdfb9238461cae.png

2

Purple Comet 2017 MS Problem 8

The positive integer

m

is a multiple of 111, and the positive integer

n

is a multiple of 31. Their sum is 2017. Find

n - m

Purple Comet 2017 HS problem 8

Find the number of trailing zeros at the end of the base-

10

representation of the integer

525^{25^2} \cdot 252^{52^5}

2

Purple Comet 2017 MS Problem 7

Find the number of positive integers less than 100 that are divisors of 300.

Purple Comet 2017 HS problem 7

Consider an alphabetized list of all the arrangements of the letters in the word BETWEEN. Then BEEENTW would be in position

1

in the list, BEEENWT would be in position

2

in the list, and so forth. Find the position that BETWEEN would be in the list.

2

Purple Comet 2017 MS Problem 6

On a typical morning Aiden gets out of bed, goes through his morning preparation, rides the bus, and walks from the bus stop to work arriving at work 120 minutes after getting out of bed. One morning Aiden got out of bed late, so he rushed through his morning preparation getting onto the bus in half the usual time, the bus ride took 25 percent longer than usual, and he ran from the bus stop to work in half the usual time it takes him to walk arriving at work 96 minutes after he got out of bed. The next morning Aiden got out of bed extra early, leisurely went through his morning preparation taking 25 percent longer than usual to get onto the bus, his bus ride took 25 percent less time than usual, and he walked slowly from the bus stop to work taking 25 percent longer than usual. How many minutes after Aiden got out of bed did he arrive at work that day?

Purple Comet 2017 HS problem 6

For some constant

k

p(x) = 3x^2 + kx + 117

has the property that

p(1) = p(10)

p(20)

2

Purple Comet 2017 MS Problem 5

Find the greatest odd divisor of

160^3

Purple Comet 2017 HS problem 5

376

chocolate bars. Min bought some of the bars, and Max bought

41

more of the bars than Min bought. After that, the store still had three times as many chocolate bars as Min bought. Find the number of chocolate bars that Min bought.

2

Purple Comet 2017 MS Problem 4

The following diagram includes six circles with radius 4, one circle with radius 8, and one circle with radius 16. The area of the shaded region is

k\pi

k

.
https://snag.gy/zNPBx0.jpg

Purple Comet 2017 HS problem 4

Find the least positive integer

m

lcm(15,m) = lcm(42,m)

lcm(a, b)

is the least common multiple of

a

b

2

Purple Comet 2017 MS Problem 3

The Stromquist Comet is visible every 61 years. If the comet is visible in 2017, what is the next leap year when the comet will be visible?

Purple Comet 2017 HS problem 3

When Phil and Shelley stand on a scale together, the scale reads

151

pounds. When Shelley and Ryan stand on the same scale together, the scale reads

132

pounds. When Phil and Ryan stand on the same scale together, the scale reads

115

pounds. Find the number of pounds Shelley weighs.

2

Purple Comet 2017 MS Problem 2

The figure below was made by gluing together 5 non-overlapping congruent squares. The figure has area 45. Find the perimeter of the figure.
https://snag.gy/ZeKf4q.jpg

Purple Comet 2017 HS problem 2

The gure below shows a large square divided into

9

congruent smaller squares. A shaded square bounded by some of the diagonals of those smaller squares has area

14

. Find the area of the large square. https://cdn.artofproblemsolving.com/attachments/5/e/bad21be1b3993586c3860efa82ab27d340dbcb.png

2

Purple Comet 2017 MS Problem 1

Caden, Zoe, Noah, and Sophia shared a pizza. Caden ate 20 percent of the pizza. Zoe ate 50 percent more of the pizza than Caden ate. Noah ate 50 percent more of the pizza than Zoe ate, and Sophia ate the rest of the pizza. Find the percentage of the pizza that Sophia ate.

Purple Comet 2017 HS problem 1

1

and counts by threes:

1, 4, 7, 10, ...

. At the same time and at the same speed, Penny counts backwards from

2017

2017, 2012, 2007, 2002,...

. Find the one number that both Paul and Penny count at the same time.