MathDB

p1. Two robots race on the plane from

(0, 0)

(a, b)

, where

a

and

b

are positive real numbers with

a < b

. The robots move at the same constant speed. However, the first robot can only travel in directions parallel to the lines

x = 0

y = 0

, while the second robot can only travel in directions parallel to the lines

y = x

y = -x

. Both robots take the shortest possible path to

(a, b)

and arrive at the same time. Find the ratio

\frac{a}{b}

.
p2. Suppose

x + \frac{1}{x} + y + \frac{1}{y} = 12

and

x^2 + \frac{1}{x^2} + y^2 + \frac{1}{y^2} = 70

. Compute

x^3 + \frac{1}{x^3} + y^3 + \frac{1}{y^3}

.
p3. Find the largest non-negative integer

a

such that

2^a

divides

3^{2^{2018}}+ 3.

p4. Suppose

z

and

w

are complex numbers, and

|z| = |w| = z \overline{w}+\overline{z}w = 1

. Find the largest possible value of

Re(z + w)

, the real part of

z + w

.
p5. Two people,

A

and

B

, are playing a game with three piles of matches. In this game, a move consists of a player taking a positive number of matches from one of the three piles such that the number remaining in the pile is equal to the nonnegative difference of the numbers of matches in the other two piles.

A

and

B

each take turns making moves, with

A

making the first move. The last player able to make a move wins. Suppose that the three piles have

10

x

, and

30

matches. Find the largest value of

x

for which

A

does not have a winning strategy.
p6. Let

A_1A_2A_3A_4A_5A_6

be a regular hexagon with side length

1

. For

n = 1

...

6

, let

B_n

be a point on the segment

A_nA_{n+1}

chosen at random (where indices are taken mod

6

, so

A_7 = A_1

). Find the expected area of the hexagon

B_1B_2B_3B_4B_5B_6

.
p7. A termite sits at the point

(0, 0, 0)

, at the center of the octahedron

|x| + |y| + |z| \le 5

. The termite can only move a unit distance in either direction parallel to one of the

x

y

, or

z

axes: each step it takes moves it to an adjacent lattice point. How many distinct paths, consisting of

5

steps, can the termite use to reach the surface of the octahedron?
p8. Let

P(x) = x^{4037} - 3 - 8 \cdot \sum^{2018}_{n=1}3^{n-1}x^n

Find the number of roots

z

P(x)

with

|z| > 1

, counting multiplicity.
p9. How many times does

01101

appear as a not necessarily contiguous substring of

0101010101010101

? (Stated another way, how many ways can we choose digits from the second string, such that when read in order, these digits read

01101

?)
p10. A perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself. For example,

28

is a perfect number because

1 + 2 + 4 + 7 + 14 = 28

. Let

n_i

denote the ith smallest perfect number. Define

f(x) =\sum_{i|n_x}\sum_{j|n_i}\frac{1}{j}

(where

\sum_{i|n_x}

means we sum over all positive integers

i

that are divisors of

n_x

). Compute

f(2)

, given there are at least

50

perfect numbers.
p11. Let

O

be a circle with chord

AB

. The perpendicular bisector to

AB

is drawn, intersecting

O

at points

C

and

D

, and intersecting

AB

at the midpoint

E

. Finally, a circle

O'

with diameter

ED

is drawn, and intersects the chord

AD

at the point

F

. Given

EC = 12

, and

EF = 7

, compute the radius of

O

.
p12. Suppose

r

s

t

are the roots of the polynomial

x^3 - 2x + 3

. Find

\frac{1}{r^3 - 2}+\frac{1}{s^3 - 2}+\frac{1}{t^3 - 2}.

p13. Let

a_1

a_2

,...,

a_{14}

be points chosen independently at random from the interval

[0, 1]

. For

k = 1

2

...

7

, let

I_k

be the closed interval lying between

a_{2k-1}

and

a_{2k}

(from the smaller to the larger). What is the probability that the intersection of

I_1

I_2

...

I_7

is nonempty?
p14. Consider all triangles

\vartriangle ABC

with area

144\sqrt3

such that

\frac{\sin A \sin B \sin C}{ \sin A + \sin B + \sin C}=\frac14.

Over all such triangles

ABC

, what is the smallest possible perimeter?
p15. Let

N

be the number of sequences

(x_1,x_2,..., x_{2018})

of elements of

\{1, 2,..., 2019\}

, not necessarily distinct, such that

x_1 + x_2 + ...+ x_{2018}

is divisible by

2018

. Find the last three digits of

N

.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement