For a positive integer $N$, we color the positive divisors of $N$ (including $1$ and $N$) with four colors. A
coloring is called multichromatic if whenever $a$, $b$ and $\gcd(a, b)$ are pairwise distinct divisors of $N$, then
they have pairwise distinct colors. What is the maximum possible number of multichromatic colorings
a positive integer can have if it is not the power of any prime?
How many ways can one fill a $3\times 3$ square grid with nonnegative integers such that no nonzero integer
appears more than once in the same row or column and the sum of the numbers in every row and
column equals $7$?
Fred the Four-Dimensional Fluffy Sheep is walking in $4$-dimensional space. He starts at the origin. Each minute, he walks from his current position $(a_1,\ a_2,\ a_3,\ a_4)$ to some position $(x_1,\ x_2,\ x_3,\ x_4)$ with integer coordinates satisfying $$(x_1−a_1)^2+(x_2−a_2)^2+(x_3−a_3)^2+(x_4−a_4)^2 = 4$$
and $\mid (x_1 + x_2 + x_3 + x_4) − (a_1 + a_2 + a_3 + a_4)\mid = 2$.
In how many ways can Fred reach $(10, 10, 10, 10)$ after exactly $40$ minutes, if he is allowed to pass
through this point during his walk?
Consider a $2\times 3$ grid where each entry is one of $0$, $1$, and $2$. For how many such grids is the sum of
the numbers in every row and in every column a multiple of $3$? One valid grid is shown below: $$\begin{bmatrix} 1& 2 & 0\\ 2& 1 &0 \end{bmatrix}$$
Let $a$ and $b$ be five-digit palindromes (without leading zeroes) such that $a < b$ and there are no other
five-digit palindromes strictly between $a$ and $b$. What are all possible values of $b - a$? (A number is a
palindrome if it reads the same forwards and backwards in base $10$.)
A $4\times 4$ window is made out of $16$ square windowpanes. How many ways are there to stain each of the
windowpanes, red, pink, or magenta, such that each windowpane is the same color as exactly two of
its neighbors? Two different windowpanes are neighbors if they share a side.
How many ways are there for Nick to travel from $(0,\ 0)$ to $(16,\ 16)$ in the coordinate plane by moving
one unit in the positive $x$ or $y$ direction at a time, such that Nick changes direction an odd number of
times?
A bag contains nine blue marbles, ten ugly marbles, and one special marble. Ryan picks marbles
randomly from this bag with replacement until he draws the special marble. He notices that none of
the marbles he drew were ugly. Given this information, what is the expected value of the number of
total marbles he drew?
Sarah stands at $(0,\ 0)$ and Rachel stands at $(6,\ 8)$ in the Euclidean plane. Sarah can only move $1$ unit
in the positive $x$ or $y$ direction, and Rachel can only move $1$ unit in the negative $x$ or $y$ direction. Each
second, Sarah and Rachel see each other, independently pick a direction to move at the same time,
and move to their new position. Sarah catches Rachel if Sarah and Rachel are ever at the same point.
Rachel wins if she is able to get to $(0,\ 0)$ without being caught; otherwise, Sarah wins. Given that both
of them play optimally to maximize their probability of winning, what is the probability that Rachel
wins?
A tourist is learning an incorrect way to sort a permutation $(p_1,\ \cdots,\ p_n)$ of the integers $(1,\ \cdots,\ n)$. We
define a fix on two adjacent elements $p_i$ and $p_{i+1}$, to be an operation which swaps the two elements
if $p_i > p_{i+1}$, and does nothing otherwise. The tourist performs $n-1$ rounds of fixes, numbered $a = 1$, $2$, $\cdots$, $n − 1$. In round a of fixes, the tourist fixes $p_a$ and $p_{a+1}$, then $p_{a+1}$ and $p_{a+2}$, and so on,
up to $p_{n−1}$ and $p_n$. In this process, there are $(n − 1) + (n − 2) + · · · + 1 = n(n−1)
2$ total fixes performed.
How many permutations of $(1,\ \cdots ,\ 2018)$ can the tourist start with to obtain $(1,\ \cdots ,\ 2018)$ after performing these steps?
A permutation of $\{1,\ 2,\ \cdots,\ 7\}$ is chosen uniformly at random. A partition of the permutation into
contiguous blocks is correct if, when each block is sorted independently, the entire permutation becomes
sorted. For example, the permutation $(3,\ 4,\ 2,\ 1,\ 6,\ 5,\ 7)$ can be partitioned correctly into the blocks $[3,\ 4,\ 2,\ 1]$ and $[6,\ 5,\ 7]$, since when these blocks are sorted, the permutation becomes $(1,\ 2,\ 3,\ 4,\ 5,\ 6,\ 7)$.
Find the expected value of the maximum number of blocks into which the permutation can be partitioned correctly.
How many ordered sequences of $36$ digits have the property that summing the digits to get a number
and taking the last digit of the sum results in a digit which is not in our original sequence? (Digits
range from $0$ to $9$.)
Lily has a $300\times 300$ grid of squares. She now removes $100\times 100$ squares from each of the four corners
and colors each of the remaining $50000$ squares black and white. Given that no $2\times 2$ square is colored
in a checkerboard pattern, find the maximum possible number of (unordered) pairs of squares such
that one is black, one is white and the squares share an edge.
Kelvin the Frog is going to roll three fair ten-sided dice with faces labelled $0,\ 1,\ 2,\ \cdots,\ 9$. First he rolls
two dice, and finds the sum of the two rolls. Then he rolls the third die. What is the probability that
the sum of the first two rolls equals the third roll?
How many ways are there to insert $+$’s between the digits of $111111111111111$ (fifteen $1$’s) so that the
result will be a multiple of $30$?
There are $2017$ jars in a row on a table, initially empty. Each day, a nice man picks ten consecutive
jars and deposits one coin in each of the ten jars. Later, Kelvin the Frog comes back to see that $N$ of
the jars all contain the same positive integer number of coins (i.e. there is an integer $d > 0$ such that $N$ of the jars have exactly $d$ coins). What is the maximum possible value of $N$?
Sam spends his days walking around the following $2\times 2$ grid of squares. $$\begin{bmatrix} 1& 2 \\ 4&3 \end{bmatrix}$$
Say that two squares are adjacent if they share a side. He starts at the square labeled $1$ and every
second walks to an adjacent square. How many paths can Sam take so that the sum of the numbers
on every square he visits in his path is equal to $20$ (not counting the square he started on)?
Kelvin the Frog likes numbers whose digits strictly decrease, but numbers that violate this condition
in at most one place are good enough. In other words, if $d_i$ denotes the $i^{th}$ digit, then $d_i\le d_{i+1}$ for
at most one value of $i$. For example, Kelvin likes the numbers $43210$, $132$, and $3$, but not the numbers $1337$ and $123$. How many $5$-digit numbers does Kelvin like?
Emily starts with an empty bucket. Every second, she either adds a stone to the bucket or removes a
stone from the bucket, each with probability $\frac{1}{2}$ . If she wants to remove a stone from the bucket and
the bucket is currently empty, she merely does nothing for that second (still with probability $\frac{1}{2}$). What
is the probability that after $2017$ seconds her bucket contains exactly $1337$ stones?
There are $2017$ frogs and $2017$ toads in a room. Each frog is friends with exactly $2$ distinct toads. Let $N$ be the number of ways to pair every frog with a toad who is its friend, so that no toad is paired
with more than one frog. Let $D$ be the number of distinct possible values of $N$, and let $S$ be the sum
of all possible values of $N$. Find the ordered pair $(D,\ S)$.
Kelvin and $15$ other frogs are in a meeting, for a total of $16$ frogs. During the meeting, each pair of
distinct frogs becomes friends with probability $\frac{1}{2}$ . Kelvin thinks the situation after the meeting is cool
if for each of the $16$ frogs, the number of friends they made during the meeting is a multiple of $4$. Find the probability of the situation being cool.
Let m be a positive integer, and let $\mathbb{T}$ denote the set of all subsets of $\{1,\ 2,\ \cdots,\ m\}$. Call a subset $\mathbb{S}$ of $\mathbb{T}$ ${\delta}-good$ if for all $s_1,\ s_2\in\mathbb{S}$, $s_1\ne s_2$, $\mid\Delta(s_1,\ s_2)\mid\ge{\delta}m$, where $\Delta$ denotes symmetric difference (the
symmetric difference of two sets is the set of elements that is in exactly one of the two sets). Find the
largest possible integer $s$ such that there exists an integer $m$ and a $\frac{1024}{2047}-good$ set of size $s$.
Compute the number of possible words $w = w_1w_2\cdots w_{100}$ satisfying: