Find the number of positive integers $j\leq 3^{2013}$ such that \[j=\sum_{k=0}^m\left((-1)^k\cdot 3^{a_k}\right)\] for some strictly increasing sequence of nonnegative integers $\{a_k\}$. For example, we may write $3=3^1$ and $55=3^0-3^3+3^4$, but $4$ cannot be written in this form.

Let $H$ be a regular hexagon of side length $x$. Call a hexagon in the same plane a "distortion" of $H$ if and only if it can be obtained from $H$ by translating each vertex of $H$ by a distance strictly less than $1$. Determine the smallest value of $x$ for which every distortion of $H$ is necessarily convex.

Quadrilateral $ABCD$ satisfies $AB = 8, BC = 5, CD = 17, DA = 10$. Let $E$ be the intersection of $AC$ and $BD$. Suppose $BE : ED = 1 : 2$. Find the area of $ABCD$.

How many polynomials $P$ with integer coefficients and degree at most $5$ satisfy $0 \le P(x) < 120$ for all $x \in \{0,1,2,3,4,5\}$?

For any positive integer $n$, $S_{n}$ be the set of all permutations of $\{1,2,3,\dots,n\}$. For each permutation $\pi \in S_n$, let $f(\pi)$ be the number of ordered pairs $(j,k)$ for which $\pi(j)>\pi(k)$ and $1\leq j<k \leq n$. Further define $g(\pi)$ to be the number of positive integers $k \leq n$ such that $\pi(k)\equiv k \pm 1 \pmod{n}$. Compute \[ \sum_{\pi \in S_{999}} (-1)^{f(\pi)+g(\pi)}. \]

Find the sum of all real numbers $x$ such that $5x^4-10x^3+10x^2-5x-11=0$.

Find the number of triples of sets $(A, B, C)$ such that: (a) $A, B, C \subseteq \{1, 2, 3, \dots , 8 \}$. (b) $|A \cap B| = |B \cap C| = |C \cap A| = 2$. (c) $|A| = |B| = |C| = 4$. Here, $|S|$ denotes the number of elements in the set $S$.

Let $ABC$ be a triangle with $AB = 13$, $BC = 14$, $CA = 15$. Let $H$ be the orthocenter of $ABC$. Find the radius of the circle with nonzero radius tangent to the circumcircles of $AHB$, $BHC$, $CHA$.

Let $m,n$ be positive integers with $m \ge n$. Let $S$ be the set of pairs $(a,b)$ of relatively prime positive integers such that $a,b \le m$ and $a+b > m$. For each pair $(a,b)\in S$, consider the nonnegative integer solution $(u,v)$ to the equation $au - bv = n$ chosen with $v \ge 0$ minimal, and let $I(a,b)$ denote the (open) interval $(v/a, u/b)$. Prove that $I(a,b) \subseteq (0,1)$ for every $(a,b)\in S$, and that any fixed irrational number $\alpha\in(0,1)$ lies in $I(a,b)$ for exactly $n$ distinct pairs $(a,b)\in S$. [i]Victor Wang, inspired by 2013 ISL N7[/i]

Find the number of positive divisors $d$ of $15!=15\cdot 14\cdot\cdots\cdot 2\cdot 1$ such that $\gcd(d,60)=5$.

A [i]root of unity[/i] is a complex number that is a solution to $ z^n \equal{} 1$ for some positive integer $ n$. Determine the number of roots of unity that are also roots of $ z^2 \plus{} az \plus{} b \equal{} 0$ for some integers $ a$ and $ b$.

Let $ABCD$ be a quadrilateral inscribed in the unit circle such that $\angle BAD$ is $30$ degrees. Let $m$ denote the minimum value of $CP + PQ + CQ$, where $P$ and $Q$ may be any points lying along rays $AB$ and $AD$, respectively. Determine the maximum value of $m$.

Compute the value of $1^{25}+2^{24}+3^{23}+\ldots+24^2+25^1$. If your answer is $A$ and the correct answer is $C$, then your score on this problem will be $\left\lfloor25\min\left(\left(\frac AC\right)^2,\left(\frac CA\right)^2\right)\right\rfloor$.

Given $a$, $b$, and $c$ are complex numbers satisfying \[ a^2+ab+b^2=1+i \] \[ b^2+bc+c^2=-2 \] \[ c^2+ca+a^2=1, \] compute $(ab+bc+ca)^2$. (Here, $i=\sqrt{-1}$)

13. How many functions $f : \{0, 1\}^3 \to \{0, 1\}$ satisfy the property that, for all ordered triples $(a_1, a_2, a_3)$ and $(b_1, b_2, b_3)$ such that $a_i \ge b_i$ for all $i$, $f(a_1, a_2, a_3) \ge f(b_1, b_2, b_3)$? 14. The very hungry caterpillar lives on the number line. For each non-zero integer $i$, a fruit sits on the point with coordinate $i$. The caterpillar moves back and forth; whenever he reaches a point with food, he eats the food, increasing his weight by one pound, and turns around. The caterpillar moves at a speed of $2^{-w}$ units per day, where $w$ is his weight. If the caterpillar starts off at the origin, weighing zero pounds, and initially moves in the positive $x$ direction, after how many days will he weigh $10$ pounds? 15. Let $ABCD$ be an isosceles trapezoid with parallel bases $AB = 1$ and $CD = 2$ and height $1$. Find the area of the region containing all points inside $ABCD$ whose projections onto the four sides of the trapezoid lie on the segments formed by $AB,BC,CD$ and $DA$.

Suppose that there are $16$ variables $\{a_{i,j}\}_{0\leq i,j\leq 3}$, each of which may be $0$ or $1$. For how many settings of the variables $a_{i,j}$ do there exist positive reals $c_{i,j}$ such that the polynomial \[f(x,y)=\sum_{0\leq i,j\leq 3}a_{i,j}c_{i,j}x^iy^j\] $(x,y\in\mathbb{R})$ is bounded below?

You are given an unlimited supply of red, blue, and yellow cards to form a hand. Each card has a point value and your score is the sum of the point values of those cards. The point values are as follows: the value of each red card is 1, the value of each blue card is equal to twice the number of red cards, and the value of each yellow card is equal to three times the number of blue cards. What is the maximum score you can get with fifteen cards?

Let $\mathbb{N} = \{1, 2, 3, \dots\}$ be the set of all positive integers, and let $f$ be a bijection from $\mathbb{N}$ to $\mathbb{N}$. Must there exist some positive integer $n$ such that $(f(1), f(2), \dots, f(n))$ is a permutation of $(1, 2, \dots, n)$?

Let $AB$ be a line segment with length 2, and $S$ be the set of points $P$ on the plane such that there exists point $X$ on segment $AB$ with $AX = 2PX$. Find the area of $S$.

Find the value of \[\sum_{a = 1}^{\infty} \sum_{b = 1}^{\infty} \sum_{c = 1}^{\infty} \frac{ab(3a + c)}{4^{a+b+c} (a+b)(b+c)(c+a)}.\]

What is the largest $n$ such that $n! + 1$ is a square?

Yannick is playing a game with $100$ rounds, starting with $1$ coin. During each round, there is an $n\%$ chance that he gains an extra coin, where $n$ is the number of coins he has at the beginning of the round. What is the expected number of coins he will have at the end of the game?

If $a$ and $b$ satisfy the equations $a +\frac1b=4$ and $\frac1a+b=\frac{16}{15}$, determine the product of all possible values of $ab$.

DeAndre Jordan shoots free throws that are worth $1$ point each. He makes $40\%$ of his shots. If he takes two shots find the probability that he scores at least $1$ point.

Let the sequence $a_i$ be defined as $a_{i+1} = 2^{a_i}$. Find the number of integers $1 \le n \le 1000$ such that if $a_0 = n$, then $100$ divides $a_{1000} - a_1$.