Let $\gamma_1, \gamma_2, \gamma_3$ be mutually externally tangent circles and $\Gamma_1, \Gamma_2, \Gamma_3$ also be mutually externally tangent circles. For each $1 \le i \le 3$, $\gamma_i$ and $\Gamma_{i+1}$ are externally tangent at $A_i$, $\gamma_i$ and $\Gamma_{i+2}$ are externally tangent at $B_i$, and $\gamma_i$ and $\Gamma_i$ do not meet. Show that the six points $A_1, A_2, A_3, B_1, B_2, B_3$ lie on either a line or a circle.

There are $m \ge 2$ blue points and $n \ge 2$ red points in three-dimensional space, and no four points are coplanar. Geoff and Nazar take turns, picking one blue point and one red point and connecting the two with a straight-line segment. Assume that Geoff starts first and the one who first makes a cycle wins. Who has the winning strategy?

Do there exist polynomials $f(x)$, $g(x)$ with real coefficients and a positive integer $k$ satisfying the following condition? (Here, the equation $x^2 = 0$ is considered to have $1$ distinct real roots. The equation $0 = 0$ has infinitely many distinct real roots.) For any real numbers $a, b$ with $(a,b) \neq (0,0)$, the number of distinct real roots of $a f(x) + b g(x) = 0$ is $k$.

For a point $P$ on the plane, denote by $\lVert P \rVert$ the distance to its nearest lattice point. Prove that there exists a real number $L > 0$ satisfying the following condition: For every $\ell > L$, there exists an equilateral triangle $ABC$ with side-length $\ell$ and $\lVert A \rVert, \lVert B \rVert, \lVert C \rVert < 10^{-2017}$.

Let $f : \mathbb{Z} \to \mathbb{R}$ be a function satisfying $f(x) + f(y) + f(z) \ge 0$ for all integers $x, y, z$ with $x + y + z = 0$. Prove that \[ f(-2017) + f(-2016) + \cdots + f(2016) + f(2017) \ge 0. \]

Find all functions $f : \mathbb{N} \to \mathbb{N}$ satisfying the following conditions: For every $n \in \mathbb{N}$, $f^{(n)}(n) = n$. (Here $f^{(1)} = f$ and $f^{(k)} = f^{(k-1)} \circ f$.) For every $m, n \in \mathbb{N}$, $\lvert f(mn) - f(m) f(n) \rvert < 2017$.

Let $\triangle ABC$ be a triangle with $\angle A \neq 60^\circ$. Let $I_B, I_C$ be the $B, C$-excenters of triangle $ABC$, let $B^\prime$ be the reflection of $B$ with respect to $AC$, and let $C^\prime$ be the reflection of $C$ with respect to $AB$. Let $P$ be the intersection of $I_C B^\prime$ and $I_B C^\prime$. Denote by $P_A, P_B, P_C$ the reflections of the point $P$ with respect to $BC, CA, AB$. Show that the three lines $A P_A, B P_B, C P_C$ meet at a single point.

For a nonzero integer $k$, denote by $\nu_2(k)$ the maximal nonnegative integer $t$ such that $2^t \mid k$. Given are $n (\ge 2)$ pairwise distinct integers $a_1, a_2, \ldots, a_n$. Show that there exists an integer $x$, distinct from $a_1, \ldots, a_n$, such that among $\nu_2(x - a_1), \ldots, \nu_2(x - a_n)$ there are at least $n/4$ odd numbers and at least $n/4$ even numbers.

For every positive integers $n,m$, show that there exist two sets $A,B$ which satisfy the following. $A$ is a set of $n$ successive positive integers, and $B$ is a set of $m$ successive positive integers. $A\cup B = \phi$ For every $a\in A$ and $b\in B$, $a$ and $b$ are not relatively prime.

Alice and Bob play a game. There are $100$ gold coins, $100$ silver coins, and $100$ bronze coins. Players take turns to take at least one coin, but they cannot take two or more coins of same kind at once. Alice goes first. The player who cannot take any coin loses. Who has a winning strategy?

Let $a,b,c,d$ be the area of four faces of a tetrahedron, satisfying $a+b+c+d=1$. Show that $$\sqrt[n]{a^n+b^n+c^n}+\sqrt[n]{b^n+c^n+d^n}+\sqrt[n]{c^n+d^n+a^n}+\sqrt[n]{d^n+a^n+b^n} \le 1+\sqrt[n]{2}$$holds for all positive integers $n$.

Find all prime number $p$ such that the number of positive integer pair $(x,y)$ satisfy the following is not $29$. $1\le x,y\le 29$ $29\mid y^2-x^p-26$

$ABC$ is an obtuse triangle satisfying $\angle A>90^\circ$, and its circumcenter $O$ and circumcircle $\omega_1$. Let $\omega_2$ be a circle passing $C$ with center $B$. $\omega_2$ meets $BC$ at $D$. $\omega_1$ meets $AD$ and $\omega_2$ at $E$ and $F(\neq C)$, respectively. $AF$ meets $\omega_2$ at $G(\neq F)$. Prove that the intersection of $CE$ and $BG$ lies on the circumcircle of $AOB$.

For a number consists of $0$ and $1$, one can perform the following operation: change all $1$ into $100$, all $0$ into $1$. For all nonnegative integer $n$, let $A_n$ be the number obtained by performing the operation $n$ times on $1$(starts with $100,10011,10011100100,\dots$), and $a_n$ be the $n$-th digit(from the left side) of $A_n$. Prove or disprove that there exists a positive integer $m$ satisfies the following: For every positive integer $l$, there exists a positive integer $k\le m$ satisfying$$a_{l+k+1}=a_1,\ a_{l+k+2}=a_2,\ \dots,\ a_{l+k+2017}=a_{2017}$$