MathDB

Round 1
p1. Elaine creates a sequence of positive integers

\{s_n\}

. She starts with

s_1 = 2018

. For

n \ge 2

, she sets

s_n =\frac12 s_{n-1}

s_{n-1}

is even and

s_n = s_{n-1} + 1

s_{n-1}

is odd. Find the smallest positive integer

n

such that

s_n = 1

, or submit “

0

” as your answer if no such

n

exists.
p2. Alice rolls a fair six-sided die with the numbers

1

through

6

, and Bob rolls a fair eight-sided die with the numbers

1

through

8

. Alice wins if her number divides Bob’s number, and Bob wins otherwise. What is the probability that Alice wins?
p3. Four circles each of radius

\frac14

are centered at the points

\left( \pm \frac14, \pm \frac14 \right)

, and ther exists a fifth circle is externally tangent to these four circles. What is the radius of this fifth circle?
Round 2
p4. If Anna rows at a constant speed, it takes her two hours to row her boat up the river (which flows at a constant rate) to Bob’s house and thirty minutes to row back home. How many minutes would it take Anna to row to Bob’s house if the river were to stop flowing?
p5. Let

a_1 = 2018

, and for

n \ge 2

define

a_n = 2018^{a_{n-1}}

. What is the ones digit of

a_{2018}

?
p6. We can write

(x + 35)^n =\sum_{i=0}^n c_ix^i

for some positive integer

n

and real numbers

c_i

. If

c_0 = c_2

, what is

n

?
Round 3
p7. How many positive integers are factors of

12!

but not of

(7!)^2

?
p8. How many ordered pairs

(f(x), g(x))

of polynomials of degree at least

1

with integer coefficients satisfy

f(x)g(x) = 50x^6 - 3200

?
p9. On a math test, Alice, Bob, and Carol are each equally likely to receive any integer score between

1

and

10

(inclusive). What is the probability that the average of their three scores is an integer?
Round 4
p10. Find the largest positive integer N such that

(a-b)(a-c)(a-d)(a-e)(b-c)(b-d)(b-e)(c-d)(c-e)(d-e)

is divisible by

N

for all choices of positive integers

a > b > c > d > e

.
p11. Let

ABCDE

be a square pyramid with

ABCD

a square and E the apex of the pyramid. Each side length of

ABCDE

6

. Let

ABCDD'C'B'A'

be a cube, where

AA'

BB'

CC'

DD'

are edges of the cube. Andy the ant is on the surface of

EABCDD'C'B'A'

at the center of triangle

ABE

(call this point

G

) and wants to crawl on the surface of the cube to

D'

. What is the length the shortest path from

G

D'

? Write your answer in the form

\sqrt{a + b\sqrt3}

, where

a

and

b

are positive integers.
p12. A six-digit palindrome is a positive integer between

100, 000

and

999, 999

(inclusive) which is the same read forwards and backwards in base ten. How many composite six-digit palindromes are there?
PS. You should use hide for answers. Rounds 5-7 have been posted [url=https://artofproblemsolving.com/community/c4h2784943p24473026]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement