Let $$f(x)=\frac{2016^x}{2016^x+\sqrt{2016}}.$$ Evaluate $$\sum_{k=0}^{2016}f\left(\frac{k}{2016}\right).$$

Simplify $\sqrt{7+\sqrt{33}}-\sqrt{7-\sqrt{33}}$.

Given a triangle $ABC$ with integer side lengths, where $BD$ is an angle bisector of $\angle ABC$, $AD=4$, $DC=6$, and $D$ is on $AC$, compute the minimum possible perimeter of $\triangle ABC$.

Compute the smallest positive integer $x$ which satisfies $x^2-8x+1\equiv0\pmod{22}$ and $x^2-22x+1\equiv0\pmod{8}$.

Given that $x$ and $y$ are real numbers, compute the minimum value of $$x^4+4x^3+8x^2+4xy+6x+4y^2+10.$$

For integers $z$, let $\#(z)$ denote the number of integer ordered pairs $(x, y)$ that satisfy $x^2 - xy + y^2 = z$. How many integers $z$ between $0$ and $150$ inclusive satisfy $\#(z) \equiv 6$ (mod $12$)?

Find the smallest positive integer solution to the equation $2^{2^k}\equiv k\pmod{29}$.

There are $2$ ways to write $2020$ as a sum of $2$ squares: $2020 = a^2 + b^2$ and $2020 = c^2 + d^2$, where $a$, $b$, $c$, and $d$ are distinct positive integers with $a < b$ and $c < d$. Compute $a+b+c+d$.

Let $ABCDEF$ be a regular hexagon, and let $G$, $H$, $I$, $J$, $K$, and $L$ be the midpoints of sides $AB$, $BC$, $CD$, $DE$, $EF$, and $FA$, respectively. The intersection of lines $\overline{AH}$, $\overline{BI}$, $\overline{CJ}$, $\overline{DK}$, $\overline{EL}$, and $\overline{FG}$ bound a smaller regular hexagon. Find the ratio of the area of the smaller hexagon to the area of $ABCDEF$.

A segment of length $1$ is drawn such that its endpoints lie on a unit circle, dividing the circle into two parts. Compute the area of the larger region.

Suppose that $x,y,z$ are positive real numbers that satisfy \begin{align*} x+y+z&=xyz\\ \frac{x^2}{16(1+x^2)}=\frac{y^2}{25(1+y^2)}&=\frac{z^2}{36(1+z^2)}. \end{align*} Compute $$\frac{x^2(1+x^2)^2}{x^2(1+z^2)^2}.$$

Consider a round table on which $2014$ people are seated. Suppose that the person at the head of the table receives a giant plate containing all the food for supper. He then serves himself and passes the plate either right or left with equal probability. Each person, upon receiving the plate, will serve himself if necessary and similarly pass the plate either left or right with equal probability. Compute the probability that you are served last if you are seated $2$ seats away from the person at the head of the table.

Rectangle $ABCD$ has $AB=20$ and $BC=15$. $2$ circles with diameters $AB$ and $AC$ intersect again at point $E$. What is the length of $DE$?

Three roots of the quartic polynomial $f(x)=x^4+ax^3+bx+c$ are $-1$, $3$, and $5$. What is $a+b-c$?

Eddy and Moor play a game with the following rules: [list=a] [*] The game begins with a pile of $N$ stones, where $N$ is a positive integer. [/*] [*] The $2$ players alternate taking turns (e.g. Eddy moves, then Moor moves, then Eddy moves, and so on). [/*] [*] During a player's turn, given $a$ stones remaining in the pile, the player may remove $b$ stones from the pile, where $\gcd(a,b)=1$ and $b\leq a$. [/*] [*] If a player cannot make a move, they lose. [/*] [/list] For example, if Eddy goes first and $N=4$, then Eddy can remove $3$ stones from the pile (since $3\leq4$ and $\gcd(3,4)=1$), leaving $1$ stone in the pile. Moor can then remove $1$ stone from the pile (since $1\leq1$ and $\gcd(1,1)=1$), leaving $0$ stones in the pile. Since Eddy cannot remove stones from an empty pile, he cannot make a move, and therefore loses. Both Eddy and Moor want to win, so they will both always make the best possible move. If Eddy moves first, for how many values of $N<2016$ can Eddy win no matter what moves Moor chooses?

Let $f(x)=\tfrac{x+a}{x+b}$ for real numbers $x$ such that $x\neq -b$. Compute all pairs of real numbers $(a,b)$ such that $f(f(x))=-\tfrac{1}{x}$ for $x\neq0$.

Let $S_n$ be the number of subsets of the first $n$ positive integers that have the same number of even values and odd values; the empty set counts as one of these subsets. Compute the smallest positive integer $n$ such that $S_n$ is a multiple of $2020$.

Let $\alpha,\beta,\gamma$ be the three real roots of the polynomial $x^3-x^2-2x+1=0$. Find all possible values of $\tfrac{\alpha}{\beta}+\tfrac{\beta}{\gamma}+\tfrac{\gamma}{\alpha}$.

Let $ABCD$ be a convex quadrilateral such that $\vartriangle ABC$ is an equilateral triangle. Let $P$ be a point inside the quadrilateral such that $\vartriangle APD$ is an equilateral triangle and $\angle PCD = 30^o$. Suppose $CP = 6$ and $CD = 8$. The area of the triangle formed by $P$, the midpoint of $\overline{BC}$, and the midpoint of $\overline{AB}$ can be expressed in simplest radical form as $\frac{a+b\sqrt{c}}{d}$ , where $a$, $b$, $c$, and $d$ are positive integers with $gcd(a, b, d) = 1$ and with $c$ not divisible by the square of any prime. Compute $a + b + c + d$.

Circles $\omega_1$ and $\omega_2$ have radii $r_1<r_2$ respectively and intersect at distinct points $X$ and $Y$. The common external tangents intersect at point $Z$. The common tangent closer to $X$ touches $\omega_1$ and $\omega_2$ at $P$ and $Q$ respectively. Line $ZX$ intersects $\omega_1$ and $\omega_2$ again at points $R$ and $S$ and lines $RP$ and $SQ$ intersect again at point $T$. If $XT=8$, $XZ=15$, and $XY=12$, then what is $\tfrac{r_1}{r_2}$?

Ash writes the positive integers from $1$ to $2016$ inclusive as a single positive integer $n=1234567891011\dots2016$. What is the result obtained by successively adding and subtracting the digits of $n$? (In other words, compute $1-2+3-4+5-6+7-8+9-1+0-1+\dots$.)

Let $\triangle ABC$ be a right triangle with hypotenuse $AC$. A square is inscribed in the triangle such that points $D,E$ are on $AC$, $F$ is on $BC$, and $G$ is on $AB$. Given that $AG=2$ and $CF=5$, what is the area of $\triangle BFG$?

Let $r = 1-\sqrt[5]{2}+ \sqrt[5]{4}-\sqrt[5]{8}+ \sqrt[5]{16}$. There exists a unique fifth-degree polynomial $P$ such that its leading coefficient is positive, all of its coefficients are integers whose greatest common factor (among all of them) is $1$, and $P(r) = 0$. Evaluate $P(10)$.

Three unit circles are inscribed inside an equilateral triangle such that each circle is tangent to each of the other $2$ circles and to $2$ sides of the triangle. Compute the area of the triangle.

Pooh has an unlimited supply of $1\times1$, $2\times2$, $3\times3$, and $4\times4$ squares. What is the minimum number of squares he needs to use in order to fully cover a $5\times5$ with no $2$ squares overlapping?