MathDB

Round 5
p13. Points

A, B, C

, and

D

lie in a plane with

AB = 6

BC = 5

, and

CD = 5

, and

AB

is perpendicular to

BC

. Point E lies on line

AD

such that

D \ne E

AE = 3

and

CE = 5

. Find

DE

.
p14. How many ordered pairs of integers

(x,y)

are solutions to

x^2y = 36 + y

?
p15. Chicken nuggets come in boxes of two sizes,

a

nuggets per box and

b

nuggets per box. We know that

899

nuggets is the largest number of nuggets we cannot obtain with some combination of

a

-sized boxes and

b

-sized boxes. How many different pairs

(a, b)

are there with

a < b

?
Round 6
p16. You are playing a game with coins with your friends Alice and Bob. When all three of you flip your respective coins, the majority side wins. For example, if Alice, Bob, and you flip Heads, Tails, Heads in that order, then you win. If Alice, Bob, and you flip Heads, Heads, Tails in that order, then you lose. Notice that more than one person will “win.” Alice and Bob design their coins as follows: a value

p

is chosen randomly and uniformly between

0

and

1

. Alice then makes a biased coin that lands on heads with probability

p

, and Bob makes a biased coin that lands on heads with probability

1 -p

. You design your own biased coin to maximize your chance of winning without knowing

p

. What is the probability that you win?
p17. There are

N

distinct students, numbered from

1

N

. Each student has exactly one hat:

y

students have yellow hats,

b

have blue hats, and

r

have red hats, where

y + b + r = N

and

y, b, r > 0

. The students stand in a line such that all the

r

people with red hats stand in front of all the

b

people with blue hats. Anyone wearing red is standing in front of everyone wearing blue. The

y

people with yellow hats can stand anywhere in the line. The number of ways for the students to stand in a line is

2016

. What is

100y + 10b + r

?
p18. Let P be a point in rectangle

ABCD

such that

\angle APC = 135^o

and

\angle BPD = 150^o

. Suppose furthermore that the distance from P to

AC

18

. Find the distance from

P

BD

.
Round 7
p19. Let triangle

ABC

be an isosceles triangle with

|AB| = |AC|

. Let

D

and

E

lie on

AB

and

AC

, respectively. Suppose

|AD| = |BC| = |EC|

and triangle

ADE

is isosceles. Find the sum of all possible values of

\angle BAC

in radians. Write your answer in the form

2 arcsin \left( \frac{a}{b}\right) + \frac{c}{d} \pi

, where

\frac{a}{b}

and

\frac{c}{d}

are in lowest terms,

-1 \le \frac{a}{b} \le 1

, and

-1 \le \frac{c}{d} \le 1

.
p20. Kevin is playing a game in which he aims to maximize his score. In the

n^{th}

round, for

n \ge 1

, a real number between

0

and

\frac{1}{3^n}

is randomly generated. At each round, Kevin can either choose to have the randomly generated number from that round as his score and end the game, or he can choose to pass on the number and continue to the next round. Once Kevin passes on a number, he CANNOT claim that number as his score. Kevin may continue playing for as many rounds as he wishes. If Kevin plays optimally, the expected value of his score is

a + b\sqrt{c}

where

a, b

, and

c

are integers and

c

is positive and not divisible by any positive perfect square other than

1

. What is

100a + 10b + c

?
p21. Lisa the ladybug (a dimensionless ladybug) lives on the coordinate plane. She begins at the origin and walks along the grid, at each step moving either right or up one unit. The path she takes ends up at

(2016, 2017)

. Define the “area” of a path as the area below the path and above the

x

-axis. The sum of areas over all paths that Lisa can take can be represented as as

a \cdot {{4033} \choose {2016}}

. What is the remainder when

a

is divided by

1000

?
PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c4h2782871p24446475]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement