There are lily pads in a row numbered $0$ to $11$, in that order. There are predators on lily pads $3$ and $6$, and a morsel of food on lily pad $10$. Fiona the frog starts on pad $0$, and from any given lily pad, has a $\frac{1}{2}$ chance to hop to the next pad, and an equal chance to jump $2$ pads. What is the probability that Fiona reaches pad $10$ without landing on either pad $3$ or pad $6$?

How many nonzero complex numbers $z$ have the property that $0$, $z$, and $z^3$ when represented by points in the complex plane, are the three distinct vertices of an equilateral triangle?

Square pyramid $ABCDE$ has base $ABCD$, which measures $3$cm on a side, and altitude $\overline{AE}$ perpendicular to the base which measures $6$cm. Point $P$ lies on $\overline{BE}$, one third of the way from $B$ to $E$; point $Q$ lies on $\overline{DE}$, one third of the way from $D$ to $E$; and point $R$ lies on $\overline{CE}$, two thirds of the way from $C$ to $E$. What is the area, in square centimeters, of $\triangle{PQR}$?

How many quadratic polynomials with real coefficients are there such that the set of roots equals the set of coefficients? (For clarification: If the polynomial is $ax^2+bc+c, a\ne 0$, and the roots are $r$ and $s$, then the requirement is that $\{a,\ b,\ c\}=\{r,\ s\}$.)

Let $\omega=-\frac{1}{2}+\frac{1}{2}i\sqrt{3}$. Let $S$ denote all points in the complex plane of the form $a+b\omega+c\omega^2$, where $0\le a\le 1$, $0\le b\le1$, and $0\le c\le 1$. What is the area of $S$?

Let $ABCD$ be a convex quadrilateral with $BC=2$ and $CD=6$. Suppose that the centroids of $\triangle{ABC}$, $\triangle{BCD}$, and $\triangle{ACD}$ form the vertices of an equilateral triangle. What is the maximum possible value of the area of $ABCD$?

Find the number of ways to divide a convex $n$-sided polygon into $(n-2)$ triangles using non-intersecting diagonals.

What is the largest factor of $130000$ that does not contain the digit $0$ or $5$?

Twenty-seven players are randomly split into three teams of nine. Given that Zack is on a different team from Mihir and Mihir is on a different team from Andrew, what is the probability that Zack and Andrew are on the same team?

A square in the $xy$-plane has area $A$, and three of its vertices have $x$-coordinates $2$, $0$, and $18$ in some order. Find the sum of all possible values of $A$.

Find the number of eight-digit positive integers that are multiples of $9$ and have all distinct digits.

Compute the smallest positive integer n for which $$\sqrt{100+\sqrt{n}}+\sqrt{100-\sqrt{n}}$$

is an integer.

Call a polygon normal if it can be inscribed in a unit circle. How many non-congruent normal polygons are there such that the square of each side length is a positive integer?

Anders is solving a math problem, and he encounters the expression $\sqrt{15!}$. He attempts to simplify this radical by expressing it as $a\sqrt{b}$ where $a$ and $b$ are positive integers. The sum of all possible distinct values of ab can be expressed in the form $q\cdot 15!$ for some rational number $q$. Find $q$.

Equilateral triangle $ABC$ has circumcircle $\omega$. Points $D$ and $E$ are chosen on minor arcs $AB$ and $AC$ of $\omega$ respectively such that $BC = DE$. Given that triangle $ABE$ has area $3$ and triangle $ACD$ has area $4$, find the area of triangle $ABC$.

$20$ players are playing in a Super Smash Bros. Melee tournament. They are ranked $1$ - $20$, and player $n$ will always beat player $m$ if $n < m$. Out of all possible tournaments where each player plays $18$ distinct other players exactly once, one is chosen uniformly at random. Find the expected number of pairs of players that win the same number of games.

Real numbers x, y, and z are chosen from the interval $[−1, 1]$ independently and uniformly at random. What is the probability that $$|x| + |y| + |z| + |x + y + z| = |x + y| + |y + z| + |z + x|$$

Given two distinct values $b_1$ and $b_2$, their product can be written in two ways: $b_1\times b_2$ and $b_2\times b_1$. Given three distinct values $b_1$, $b_2$, and $b_3$, their products can be expressed in $12$ ways: $b_1\times(b_2\times b_3)$, $(b_1\times b_2)\times b_3$, $b_1\times(b_3\times b_2)$, $(b_1\times b_3)\times b_2$, $b_2\times(b_3\times b_1)$, $(b_2\times b_3)\times b_1$, $b_2\times(b_1\times b_3)$, $(b_2\times b_1)\times b_3$, $b_3\times(b_1\times b_2)$, $(b_3\times b_1)\times b_2$, $b_3\times(b_2\times b_1)$, and $(b_3\times b_2)\times b_1$. Please note that in this definition, orders matter and parentheses etc cannot be simplified. The question is how many different ways to express the product of $n$ distinct values?

Let $x_i\in\{+1,\ -1\}$, $i=1,\ 2,\ \cdots,\ 2n$. If their sum equals $0$ and the following inequality holds for any positive integer $k$ satisfying $1\le k < 2n$: $$x_1+x_2+\cdots + x_k\ge 0$$

Find the number of possible ordered sequence $\{x_1,\ x_2,\ \cdots,\ x_{2n}\}$.

(Hanoi Tower) There are $3$ identical rods labeled as $A$, $B$, $C$; and $n$ disks of different sizes which can be slide onto any of these three rods. Initially, the $n$ disks are stacked in ascending order of their sizes on $A$. What is the minimal number of moves in order to transfer all the disks to $B$ providing that each move can only transfer one disk to another rod's topmost position and at no time, a bigger disk can be placed on top of a smaller one.

Find the total number of sequences of length $n$ containing only letters $A$ and $B$ such that no two $A$s are next to each other. For example, for $n = 2$, there are $3$ possible sequences: $AB$, $BA$, and $BB$.

How many distinct permutations of the letters of the word REDDER are there that do not contain a palindromic substring of length at least two? (A substring is a contiguous block of letters that is part of the string. A string is palindromic if it is the same when read backwards.)

Your math friend Steven rolls five fair icosahedral dice (each of which is labelled $1$, $2$, $\cdots$, $20$ on its sides). He conceals the results but tells you that at least half of the rolls are $20$. Suspicious, you examine the first two dice and find that they show $20$ and $19$ in that order. Assuming that Steven is truthful, what is the probability that all three remaining concealed dice show $20$?

Reimu and Sanae play a game using $4$ fair coins. Initially both sides of each coin are white. Starting with Reimu, they take turns to color one of the white sides either red or green. After all sides are colored, the 4 coins are tossed. If there are more red sides showing up, then Reimu wins, and if there are more green sides showing up, then Sanae wins. However, if there is an equal number of red sides and green sides, then neither of them wins. Given that both of them play optimally to maximize the probability of winning, what is the probability that Reimu wins?