MathDB

p1. Suppose Mitchell has a fair die. He is about to roll it six times. The probability that he rolls

1

2

3

4

5

, and then

6

in that order is

p

. The probability that he rolls

2

2

4

4

6

, and then

6

in that order is

q

. What is

p - q

?
p2. What is the smallest positive integer

x

such that

x \equiv 2017

(mod

2016

) and

x \equiv 2016

(mod

2017

) ?
p3. The vertices of triangle

ABC

lie on a circle with center

O

. Suppose the measure of angle

ACB

45^o

. If

|AB| = 10

, then what is the distance between

O

and the line

AB

?
p4. A “word“ is a sequence of letters such as

YALE

and

AELY

. How many distinct

3

-letter words can be made from the letters in

BOOLABOOLA

where each letter is used no more times than the number of times it appears in

BOOLABOOLA

?
p5. How many distinct complex roots does the polynomial

p(x) = x^{12} - x^8 - x^4 + 1

have?
p6. Notice that

1 = \frac12 + \frac13 + \frac16

, that is,

1

can be expressed as the sum of the three fractions

\frac12

\frac13

, and

\frac16

, where each fraction is in the form

\frac{1}{n}

, with each

n

different. Give a

6

-tuple of distinct positive integers

(a, b, c, d, e, f)

where

a < b < c < d < e < f

such that

\frac{1}{a} +\frac{1}{b} + \frac{1}{c} + \frac{1}{d} + \frac{1}{e} + \frac{1}{f} = 1

and explain how you arrived at your

6

-tuple. Multiple answers will be accepted.
p7. You have a Monopoly board, an

11 \times 11

square grid with the

9 \times 9

internal square grid removed, where every square is blank except for Go, which is the square in the bottom right corner. During your turn, you determine how many steps forward (which is in the counterclockwise direction) to move by rolling two standard

6

-sided dice. Let

S

be the set of squares on the board such that if you are initially on a square in

S

, no matter what you roll with the dice, you will always either land on Go (move forward enough squares such that you end up on Go) or you pass Go (you move forward enough squares such that you step on Go during your move and then you advance past Go). You randomly and uniformly select one square in

S

as your starting position. What is the probability that you land on Go?
p8. Using

L

-shaped triominos, and dominos, where each square of a triomino and a domino covers one unit, what is the minimum number of tiles needed to cover a

3

-by-

2017

rectangle without any gaps?
p9. Does there exist a pair of positive integers

(x, y)

, where

x < y

, such that

x^2 + y^2 = 1009^3

? If so, give a pair

(x, y)

and explain how you found that pair. If not, explain why.
p10. Triangle

ABC

has inradius

8

and circumradius

20

. Let

M

be the midpoint of side

BC

, and let

N

be the midpoint of arc

BC

on the circumcircle not containing

A

. Let

s_A

denote the length of segment

MN

, and define

s_B

and

s_C

similarly with respect to sides

CA

and

AB

. Evaluate the product

s_As_Bs_C

.
p11. Julia and Dan want to divide up

256

dollars in the following way: in the first round, Julia will offer Dan some amount of money, and Dan can choose to accept or reject the offer. If Dan accepts, the game is over. Otherwise, if Dan rejects, half of the money disappears. In the second round, Dan can offer Julia part of the remaining money. Julia can then choose to accept or reject the offer. This process goes on until an offer is accepted or until

4

rejections have been made; once

4

rejections are made, all of the money will disappear, and the bargaining process ends. If Julia or Dan is indifferent between accepting and rejecting an offer, they will accept the offer. Given that Julia and Dan are both rational and both have the goal of maximizing the amount of money they get, how much will Julia offer Dan in the first round?
p12. A perfect partition of a positive integer

N

is an unordered set of numbers (where numbers can be repeated) that sum to

N

with the property that there is a unique way to express each positive integer less than

N

as a sum of elements of the set. Repetitions of elements of the set are considered identical for the purpose of uniqueness. For example, the only perfect partitions of

3

are

\{1, 1, 1\}

and

\{1, 2\}

\{1, 1, 3, 4\}

is NOT a perfect partition of

9

because the sum

4

can be achieved in two different ways:

4

and

1 + 3

. How many integers

1 \le N \le 40

each have exactly one perfect partition?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement