In rectangle $ABCD$, points $E$ and $F$ lie on sides $AB$ and $CD$ respectively such that both $AF$ and $CE$ are perpendicular to diagonal $BD$. Given that $BF$ and $DE$ separate $ABCD$ into three polygons with equal area, and that $EF = 1$, find the length of $BD$.

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 $m$.

Let $P$ be the number of ways to partition $2013$ into an ordered tuple of prime numbers. What is $\log_2 (P)$? If your answer is $A$ and the correct answer is $C$, then your score on this problem will be $\left\lfloor\frac{125}2\left(\min\left(\frac CA,\frac AC\right)-\frac35\right)\right\rfloor$ or zero, whichever is larger.

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

22. Let the function $f : \mathbb{Z} \to \mathbb{Z}$ take only integer inputs and have integer outputs. For any integers $x$ and $y$, f satisfies $$f(x) + f(y) = f(x + 1) + f(y - 1)$$ If $f(2016) = 6102$ and $f(6102) = 2016$, what is $f(1)$? 23. Let $d$ be a randomly chosen divisor of $2016$. Find the expected value of $$\frac{d^2}{d^2 + 2016}$$ 24. Consider an infinite grid of equilateral triangles. Each edge (that is, each side of a small triangle) is colored one of $N$ colors. The coloring is done in such a way that any path between any two nonadjecent vertices consists of edges with at least two different colors. What is the smallest possible value of $N$?

Let $N = 2^{\left(2^2\right)}$ and $x$ be a real number such that $N^{\left(N^N\right)} = 2^{(2^x)}$. Find $x$.

For all real numbers $x$, let \[ f(x) = \frac{1}{\sqrt[2011]{1-x^{2011}}}. \] Evaluate $(f(f(\ldots(f(2011))\ldots)))^{2011}$, where $f$ is applied $2010$ times.

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

19. Let $S$ be the set of all positive integers whose prime factorizations only contain powers of the primes $2$ and $2017$ ($1$, powers of $2$, and powers of $2017$ are thus contained in $S$). Compute $\sum_{s\in S}\frac1s$. 20. Let $\mathcal{V}$ be the volume enclosed by the graph $$x^ {2016} + y^{2016} + z^2 = 2016$$ Find $\mathcal{V}$ rounded to the nearest multiple of ten. 21. Zlatan has $2017$ socks of various colours. He wants to proudly display one sock of each of the colours, and he counts that there are $N$ ways to select socks from his collection for display. Given this information, what is the maximum value of $N$?

How many positive integers $k$ are there such that \[\dfrac k{2013}(a+b)=lcm(a,b)\] has a solution in positive integers $(a,b)$?

Let \[ A = \frac{1}{6}((\log_2(3))^3-(\log_2(6))^3-(\log_2(12))^3+(\log_2(24))^3) \]. Compute $2^A$.

Contessa is taking a random lattice walk in the plane, starting at $(1,1)$. (In a random lattice walk, one moves up, down, left, or right $1$ unit with equal probability at each step.) If she lands on a point of the form $(6m,6n)$ for $m,n \in \mathbb{Z}$, she ascends to heaven, but if she lands on a point of the form $(6m+3,6n+3)$ for $m,n \in \mathbb{Z}$, she descends to hell. What is the probability she ascends to heaven?

Let $W$ be the hypercube $\{(x_1,x_2,x_3,x_4)\,|\,0\leq x_1,x_2,x_3,x_4\leq 1\}$. The intersection of $W$ and a hyperplane parallel to $x_1+x_2+x_3+x_4=0$ is a non-degenerate $3$-dimensional polyhedron. What is the maximum number of faces of this polyhedron?

Let $ABCD$ be a parallelogram. Points $X$ and $Y$ lie on segments $AB$ and $AD$ respectively, and $AC$ intersects $XY$ at point $Z$. Prove that \[\frac{AB}{AX} + \frac{AD}{AY} = \frac{AC}{AZ}.\]

Let $\triangle ABC$ be a triangle with $\angle BAC>90^{\circ}$, $AB=5$ and $AC=7$. Points $D$ and $E$ lie on segment $BC$ such that $BD=DE=EC$. If $\angle BAC+\angle DAE=180^{\circ}$, compute $BC$.

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 $ABCD$.

A convex polygon on the plane is called [i]wide[/i] if the projection of the polygon onto any line in the same plane is a segment with length at least 1. Prove that a circle of radius $\tfrac{1}{3}$ can be placed completely inside any wide polygon.

A point $P$ lies at the center of square $ABCD$. A sequence of points $\{P_n\}$ is determined by $P_0 = P$, and given point $P_i$, point $P_{i+1}$ is obtained by reflecting $P_i$ over one of the four lines $AB$, $BC$, $CD$, $DA$, chosen uniformly at random and independently for each $i$. What is the probability that $P_8 = P$?

Triangle $ABC$ has incenter $I$. Let $D$ be the foot of the perpendicular from $A$ to side $BC$. Let $X$ be a point such that segment $AX$ is a diameter of the circumcircle of triangle $ABC$. Given that $ID = 2$, $IA = 3$, and $IX = 4$, compute the inradius of triangle $ABC$.

25. Chris and Paul each rent a different room of a hotel from rooms $1 - 60$. However, the hotel manager mistakes them for one person and gives ”Chris Paul” a room with Chris’s and Paul’s room concatenated. For example, if Chris had $15$ and Paul had $9$, ”Chris Paul” has $159$. If there are $360$ rooms in the hotel, what is the probability that ”Chris Paul” has a valid room? 26. Find the number of ways to choose two nonempty subsets $X$ and $Y$ of $\{1, 2, \ldots , 2001\}$, such that $|Y| = 1001$ and the smallest element of $Y$ is equal to the largest element of $X$. 27. Let $r_1, r_2, r_3, r_4$ be the four roots of the polynomial $x^4 - 4x^3 + 8x^2 - 7x + 3$. Find the value of $$\frac{r_1^2}{r_2^2+r_3^2+r_4^2}+\frac{r_2^2}{r_1^2+r_3^2+r_4^2}+\frac{r_3^2}{r_1^2+r_2^2+r_4^2}+\frac{r_4^2}{r_1^2+r_2^2+r_3^2}$$

The sequence $(z_n)$ of complex numbers satisfies the following properties: [list] [*]$z_1$ and $z_2$ are not real. [*]$z_{n+2}=z_{n+1}^2z_n$ for all integers $n\geq 1$. [*]$\dfrac{z_{n+3}}{z_n^2}$ is real for all integers $n\geq 1$. [*]$\left|\dfrac{z_3}{z_4}\right|=\left|\dfrac{z_4}{z_5}\right|=2$. [/list] Find the product of all possible values of $z_1$.

Let $ABCDEF$ be a regular hexagon, and let $P$ be a point inside quadrilateral $ABCD$. If the area of triangle $PBC$ is $20$, and the area of triangle $PAD$ is $23$, compute the area of hexagon $ABCDEF$.

Let $ABC$ be an acute triangle with circumcenter $O$, orthocenter $H$, and circumcircle $\Omega$. Let $M$ be the midpoint of $AH$ and $N$ the midpoint of $BH$. Assume the points $M$, $N$, $O$, $H$ are distinct and lie on a circle $\omega$. Prove that the circles $\omega$ and $\Omega$ are internally tangent to each other. [i]Dhroova Aiylam and Evan Chen[/i]

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.

The polynomial $f(x)=x^3-3x^2-4x+4$ has three real roots $r_1$, $r_2$, and $r_3$. Let $g(x)=x^3+ax^2+bx+c$ be the polynomial which has roots $s_1$, $s_2$, and $s_3$, where $s_1=r_1+r_2z+r_3z^2$, $s_2=r_1z+r_2z^2+r_3$, $s_3=r_1z^2+r_2+r_3z$, and $z=\frac{-1+i\sqrt3}2$. Find the real part of the sum of the coefficients of $g(x)$.