2018 Purple Comet Problems

Part of Purple Comet Problems

Subcontests

(30)

1

Purple Comet 2018 HS problem 30

One right pyramid has a base that is a regular hexagon with side length

1

, and the height of the pyramid is

8

. Two other right pyramids have bases that are regular hexagons with side length

4

, and the heights of those pyramids are both

7

. The three pyramids sit on a plane so that their bases are adjacent to each other and meet at a single common vertex. A sphere with radius

4

rests above the plane supported by these three pyramids. The distance that the center of the sphere is from the plane can be written as

\frac{p\sqrt{q}}{r}

p, q

r

are relatively prime positive integers, and

q

is not divisible by the square of any prime. Find

p+q+r

1

Purple Comet 2018 HS problem 29

Find the three-digit positive integer

n

\binom n3 \binom n4 \binom n5 \binom n6

is a perfect square.

1

Purple Comet 2018 HS problem 28

\vartriangle ABC

D, E

F

\overline{BC}

\overline{AD}

is an angle bisector of

\angle BAC

\overline{AE}

is a median, and

\overline{AF}

is an altitude. Given that

AB = 154

AC = 128

9 \times DE = EF,

find the side length

BC

1

Purple Comet 2018 HS problem 27

p < q < r < s

are prime numbers such that

pqrs + 1 = 4^{p+q}

r + s

1

Purple Comet 2018 HS problem 26

a, b

c

be real numbers. Let

u = a^2 + b^2 + c^2

v = 2ab + 2bc + 2ca

2018u = 1001v + 1024

. Find the maximum possible value of

35a - 28b - 3c

1

Purple Comet 2018 HS problem 25

If a and b are in the interval

\left(0, \frac{\pi}{2}\right)

13(\sin a + \sin b) + 43(\cos a + \cos b) = 2\sqrt{2018}

\tan a + \tan b = \frac{m}{n}

m

n

are relatively prime positive integers. Find

m + n

1

Purple Comet 2018 HS problem 24

Five girls and five boys randomly sit in ten seats that are equally spaced around a circle. The probability that there is at least one diameter of the circle with two girls sitting on opposite ends of the diameter is

\frac{m}{n}

m

n

are relatively prime positive integers. Find

m + n

1

Purple Comet 2018 HS problem 23

a, b

c

be integers simultaneously satisfying the equations

4abc + a + b + c = 2018

ab + bc + ca = -507

|a| + |b|+ |c|

1

Purple Comet 2018 HS problem 22

Positive integers

a

b

a^3 + 32b + 2c = 2018

b^3 + 32a + 2c = 1115

a^2 + b^2 + c^2

1

Purple Comet 2018 HS problem 21

x

be in the interval

\left(0, \frac{\pi}{2}\right)

\sin x - \cos x = \frac12

\sin^3 x + \cos^3 x = \frac{m\sqrt{p}}{n}

m, n

p

are relatively prime positive integers, and

p

is not divisible by the square of any prime. Find

m + n + p

2

Purple Comet 2018 MS problem 20

ABCD

be a square with side length

6

X, Y

Z

are congruent circles with centers inside the square such that

X

is tangent to both sides

\overline{AB}

\overline{AD}

Y

is tangent to both sides

\overline{AB}

\overline{BC}

Z

is tangent to side

\overline{CD}

and both circles

X

Y

. The radius of the circle

X

m -\sqrt{n}

m

n

are positive integers. Find

m + n

Purple Comet 2018 HS problem 20

Aileen plays badminton where she and her opponent stand on opposite sides of a net and attempt to bat a birdie back and forth over the net. A player wins a point if their opponent fails to bat the birdie over the net. When Aileen is the server (the first player to try to hit the birdie over the net), she wins a point with probability

\frac{9}{10}

. Each time Aileen successfully bats the birdie over the net, her opponent, independent of all previous hits, returns the birdie with probability

\frac{3}{4}

. Each time Aileen bats the birdie, independent of all previous hits, she returns the birdie with probability

\frac{m}{n}

m

n

are relatively prime positive integers. Find

m + n

2

Purple Comet 2018 MS problem 19

Two identical blue blocks, two identical red blocks, two identical green blocks, and two identical purple blocks are placed next to each other in a row. Find the number of distinct arrangements of these blocks where no blue block is placed next to a red block, and no green block is placed next to a purple block.

Purple Comet 2018 HS problem 19

a

b

are positive real numbers such that

3\log_{101}\left(\frac{1,030,301-a-b}{3ab}\right) = 3 - 2 \log_{101}(ab)

101 - \sqrt[3]{a}- \sqrt[3]{b}

2

Purple Comet 2018 MS problem 18

ABCD

has side lengths

AB = 6\sqrt3

BC = 8\sqrt3

. The probability that a randomly chosen point inside the rectangle is closer to the diagonal

\overline{AC}

than to the outside of the rectangle is

\frac{m}{n}

m

n

are relatively prime positive integers. Find

m + n

Purple Comet 2018 HS problem 18

Find the positive integer

k

such that the roots of

x^3 - 15x^2 + kx -1105

are three distinct collinear points in the complex plane.

2

Purple Comet 2018 MS problem 17

a, b, c

d

be real numbers such that

a^2 + b^2 + c^2 + d^2 = 3a + 8b + 24c + 37d = 2018

3b + 8c + 24d + 37a

Purple Comet 2018 HS problem 17

One afternoon a bakery finds that it has

300

cups of flour and

300

cups of sugar on hand. Annie and Sam decide to use this to make and sell some batches of cookies and some cakes. Each batch of cookies will require

1

cup of flour and

3

cups of sugar. Each cake will require

2

cups of flour and

1

cup of sugar. Annie thinks that each batch of cookies should sell for

2

dollars and each cake for

1

dollar, but Sam thinks that each batch of cookies should sell for

1

dollar and each cake should sell for

3

dollars. Find the difference between the maximum dollars of income they can receive if they use Sam's selling plan and the maximum dollars of income they can receive if they use Annie's selling plan.

2

Purple Comet 2018 MS problem 16

\vartriangle ABC

D

be a point on side

\overline{AB}

F

be a point on side

\overline{AC}

E

be a point inside the triangle so that

\overline{DE}\parallel \overline{AC}

\overline{EF} \parallel \overline{AB}

AF = 6, AC = 33, AD = 7, AB = 26

, and the area of quadrilateral

ADEF

14

, nd the area of

\vartriangle ABC

Purple Comet 2018 HS problem 16

If you roll four standard, fair six-sided dice, the top faces of the dice can show just one value (for example,

3333

), two values (for example,

2666

), three values (for example,

5215

), or four values (for example,

4236

). The mean number of values that show is

\frac{m}{n}

m

n

are relatively prime positive integers. Find

m + n

2

Purple Comet 2018 MS problem 15

There are integers

a_1, a_2, a_3,...,a_{240}

x(x + 1)(x + 2)(x + 3) ... (x + 239) =\sum_{n=1}^{240}a_nx^n

. Find the number of integers

k

1\le k \le 240

such that ak is a multiple of

3

Purple Comet 2018 HS problem 15

a

b

be real numbers such that

\frac{1}{a^2} +\frac{3}{b^2} = 2018a

\frac{3}{a^2} +\frac{1}{b^2} = 290b

\frac{ab}{b-a }= \frac{m}{n}

m

n

are relatively prime positive integers. Find

m + n

2

Purple Comet 2018 MS problem 14

Find the number of ordered quadruples of positive integers

(a,b,c, d)

ab + cd = 10

Purple Comet 2018 HS problem 14

A complex number

z

whose real and imaginary parts are integers satisfies

\left(Re(z) \right)^4 +\left(Re(z^2)\right)^2 + |z|^4 =(2018)(81)

Re(w)

Im(w)

are the real and imaginary parts of

w

, respectively. Find

\left(Im(z) \right)^2

2

Purple Comet 2018 MS problem 13

x

y

are nonzero real numbers simultaneously satisfying the equations

x + \frac{2018}{y}= 1000

\frac{9}{x}+ y = 1

. Find the maximum possible value of

x + 1000y

Purple Comet 2018 HS problem 13

Five lighthouses are located, in order, at points

A, B, C, D

E

along the shore of a circular lake with a diameter of

10

miles. Segments

AD

BE

are diameters of the circle. At night, when sitting at

A

, the lights from

B, C, D

E

appear to be equally spaced along the horizon. The perimeter in miles of pentagon

ABCDE

m +\sqrt{n}

m

n

are positive integers. Find

m + n

2

Purple Comet 2018 MS problem 12

\overline{AB}

has perpendicular bisector

\overline{CD}

C

is the midpoint of

\overline{AB}

. The segments have lengths

AB = 72

CD = 60

R

be the set of points

P

that are midpoints of line segments

\overline{XY}

X

\overline{AB}

Y

\overline{CD}

. Find the area of the region

R

Purple Comet 2018 HS problem 12

A jeweler can get an alloy that is

40\%

200

dollars per ounce, an alloy that is

60\%

300

dollar per ounce, and an alloy that is

90\%

400

dollars per ounce. The jeweler will purchase some of these gold alloy products, melt them down, and combine them to get an alloy that is

50\%

gold. Find the minimum number of dollars the jeweler will need to spend for each ounce of the alloy she makes.

2

Purple Comet 2018 MS problem 11

Find the number of positive integers less than

2018

that are divisible by

6

but are not divisible by at least one of the numbers

4

9

Purple Comet 2018 HS problem 11

Find the number of positive integers

k \le 2018

for which there exist integers

m

n

k = 2^m + 2^n

64 = 2^5 + 2^5

65 = 2^0 + 2^6

66 = 2^1 + 2^6

2

Purple Comet 2018 MS problem 10

The triangle below is divided into nine stripes of equal width each parallel to the base of the triangle. The darkened stripes have a total area of

135

. Find the total area of the light colored stripes. https://cdn.artofproblemsolving.com/attachments/0/8/f34b86ccf50ef3944f5fbfd615a68607f4fadc.png

Purple Comet 2018 HS problem 10

Find the remainder when

11^{2018}

100

2

Purple Comet 2018 MS problem 9

k > 0

50x + ky = 1240

ky = 8x + 544

intersect at right angles at the point

(m,n)

m + n

Purple Comet 2018 HS problem 9

trapezoid has side lengths

10, 10, 10

22

. Each side of the trapezoid is the diameter of a semicircle with the two semicircles on the two parallel sides of the trapezoid facing outside the trapezoid and the other two semicircles facing inside the trapezoid as shown. The region bounded by these four semicircles has area

m + n\pi

, where m and n are positive integers. Find

m + n

. https://3.bp.blogspot.com/-s8BoUPKVUQk/XoEaIYvaz4I/AAAAAAAALl0/ML0klwHogGYWkNhY6maDdI93_GkfL_eyQCK4BGAYYCw/s200/2018%2Bps%2Bhs9.png

2

Purple Comet 2018 MS problem 8

AE

of regular pentagon

ABCDE

there is an equilateral triangle

AEF

AB

of the pentagon there is a square

ABHG

as shown. Find the degree measure of angle

AFG

. https://cdn.artofproblemsolving.com/attachments/7/7/0d689d2665e67c9f9afdf193fb0a2db6dddb3d.png

Purple Comet 2018 HS problem 8

a

b

be positive integers such that

2a - 9b + 18ab = 2018

b - a

2

Purple Comet 2018 MS problem 7

Bradley is driving at a constant speed. When he passes his school, he notices that in

20

minutes he will be exactly

\frac14

of the way to his destination, and in

45

minutes he will be exactly

\frac13

of the way to his destination. Find the number of minutes it takes Bradley to reach his destination from the point where he passes his school.

Purple Comet 2018 HS problem 7

10

years the product of Melanie's age and Phil's age will be

400

more than it is now. Find what the sum of Melanie's age and Phil's age will be

6

years from now.

2

Purple Comet 2018 MS problem 6

Find the greatest integer

n

10^n

\frac{2^{10^5} 5^{2^{10}}}{10^{5^2}}

Purple Comet 2018 HS problem 6

ABC

AB = AC

D

\overline{BC}

AD = CD

\angle BAD = 36^o

. Find the degree measure of

\angle BAC

2

Purple Comet 2018 MS problem 5

The positive integer

m

is a multiple of

101

, and the positive integer

n

is a multiple of

63

2018

m - n

Purple Comet 2018 HS problem 5

One afternoon at the park there were twice as many dogs as there were people, and there were twice as many people as there were snakes. The sum of the number of eyes plus the number of legs on all of these dogs, people, and snakes was

510

. Find the number of dogs that were at the park.

2

Purple Comet 2018 MS problem 4

The diagram below shows a large square with each of its sides divided into four equal segments. The shaded square whose sides are diagonals drawn to these division points has area

13

. Find the area of the large square. https://cdn.artofproblemsolving.com/attachments/8/3/bee223ef39dea493d967e7ebd5575816954031.png

Purple Comet 2018 HS problem 4

The following diagram shows a grid of

36

cells. Find the number of rectangles pictured in the diagram that contain at least three cells of the grid. https://cdn.artofproblemsolving.com/attachments/a/4/e9ba3a35204ec68c17a364ebf92cc107eb4d7a.png

2

Purple Comet 2018 MS problem 3

\left(\frac{\frac13+1}{3} +\frac{1+ \frac13}{3} \right) / \left(\frac{3}{\frac{1}{3+1}+\frac{ 1}{1+3}}\right)

can be written as

\frac{m}{n}

m

n

are relatively prime positive integers. Find

m + n

Purple Comet 2018 HS problem 3

x

so that the arithmetic mean of

x, 3x, 1000

3000

2018

2

Purple Comet 2018 MS #2

The following figure is made up of many

2

4

tiles such that adjacent tiles always share an edge of length

2

. Find the perimeter of this figure.

Purple Comet 2018 HS problem 2

A triangle with side lengths

16

18

21

has a circle with radius

6

centered at each vertex. Find

n

so that the total area inside the three circles but outside of the triangle is

n\pi

. https://4.bp.blogspot.com/-dpCi7Gai3ZE/XoEaKo3C5wI/AAAAAAAALl8/KAuCVDT9R5MiIA_uTfRyoQmohEVw9cuVACK4BGAYYCw/s200/2018%2Bpc%2Bhs2.png

2

Purple Comet 2018 MS #1

n

such that the mean of

\frac74

\frac65

\frac1n

1

Purple Comet 2018 HS problem 1

Find the positive integer

n

\frac12 \cdot \frac34 + \frac56 \cdot \frac78 + \frac{9}{10}\cdot \frac{11}{12 }= \frac{n}{1200}