Find all real numbers $x_1, \dots, x_{2016}$ that satisfy the following equation for each $1 \le i \le 2016$. (Here $x_{2017} = x_1$.) \[ x_i^2 + x_i - 1 = x_{i+1} \]

Let the incircle of triangle $ABC$ meet the sides $BC$, $CA$, $AB$ at $D$, $E$, $F$, and let the $A$-excircle meet the lines $BC$, $CA$, $AB$ at $P$, $Q$, $R$. Let the line passing through $A$ and perpendicular to $BC$ meet the lines $EF$, $QR$ at $K$, $L$. Let the intersection of $LD$ and $EF$ be $S$, and the intersection of $KP$ and $QR$ be $T$. Prove that $A$, $S$, $T$ are collinear.

Let $p > 10^9$ be a prime number such that $4p + 1$ is also prime. Prove that the decimal expansion of $\frac{1}{4p+1}$ contains all the digits $0,1, \ldots, 9$.

Two integers $0 < k < n$ and distinct real numbers $a_1, a_2, \dots ,a_n$ are given. Define the sets as the following, where all indices are modulo $n$. \begin{align*} A &= \{ 1 \le i \le n : a_i > a_{i-k}, a_{i-1}, a_{i+1}, a_{i+k} \text{ or } a_i < a_{i-k}, a_{i-1}, a_{i+1}, a_{i+k} \} \\ B &= \{ 1 \le i \le n : a_i > a_{i-k}, a_{i+k} \text{ and } a_i < a_{i-1}, a_{i+1} \} \\ C &= \{ 1 \le i \le n ; a_i > a_{i-1}, a_{i+1} \text{ and } a_i < a_{i-k}, a_{i+k} \} \end{align*}Prove that $\lvert A \rvert \ge \lvert B \rvert + \lvert C \rvert$.

Find the maximal possible $n$, where $A_1, \dots, A_n \subseteq \{1, 2, \dots, 2016\}$ satisfy the following properties. - For each $1 \le i \le n$, $\lvert A_i \rvert = 4$. - For each $1 \le i < j \le n$, $\lvert A_i \cap A_j \rvert$ is even.

A finite set $S$ of positive integers is given. Show that there is a positive integer $N$ dependent only on $S$, such that any $x_1, \dots, x_m \in S$ whose sum is a multiple of $N$, can be partitioned into groups each of whose sum is exactly $N$. (The numbers $x_1, \dots, x_m$ need not be distinct.)

A infinite sequence $\{ a_n \}_{n \ge 0}$ of real numbers satisfy $a_n \ge n^2$. Suppose that for each $i, j \ge 0$ there exist $k, l$ with $(i,j) \neq (k,l)$, $l - k = j - i$, and $a_l - a_k = a_j - a_i$. Prove that $a_n \ge (n + 2016)^2$ for some $n$.

There are distinct points $A_1, A_2, \dots, A_{2n}$ with no three collinear. Prove that one can relabel the points with the labels $B_1, \dots, B_{2n}$ so that for each $1 \le i < j \le n$ the segments $B_{2i-1} B_{2i}$ and $B_{2j-1} B_{2j}$ do not intersect and the following inequality holds. \[ B_1 B_2 + B_3 B_4 + \dots + B_{2n-1} B_{2n} \ge \frac{2}{\pi} (A_1 A_2 + A_3 A_4 + \dots + A_{2n-1} A_{2n}) \]