There are $44$ distinct holes in a line and $2017$ ants. Each ant comes out of a hole and crawls along the line with a constant speed into another hole, then comes in. Let $T$ be the set of moments for which the ant comes in or out of the holes. Given that $|T|\leq 45$ and the speeds of the ants are distinct. Prove that there exists two ants that don't collide.

For each positive integer $n$, set $x_n=\binom{2n}{n}$. a. Prove that if $\frac{2017^k}{2}<n<2017^k$ for some positive integer $k$ then $2017$ divides $x_n$. b. Find all positive integer $h>1$ such that there exists positive integers $N,T$ such that $(x_n)_{n>N}$ is periodic mod $h$ with period $T$.

Triangle $ABC$ with incircle $(I)$ touches the sides $AB, BC, AC$ at $F, D, E$, res. $I_b, I_c$ are $B$- and $C-$ excenters of $ABC$. $P, Q$ are midpoints of $I_bE, I_cF$. $(PAC)\cap AB=\{ A, R\}$, $(QAB)\cap AC=\{ A,S\}$. a. Prove that $PR, QS, AI$ are concurrent. b. $DE, DF$ cut $I_bI_c$ at $K, J$, res. $EJ\cap FK=\{ M\}$. $PE, QF$ cut $(PAC), (QAB)$ at $X, Y$ res. Prove that $BY, CX, AM$ are concurrent.

Triangle $ABC$ is inscribed in circle $(O)$. $A$ varies on $(O)$ such that $AB>BC$. $M$ is the midpoint of $AC$. The circle with diameter $BM$ intersects $(O)$ at $R$. $RM$ intersects $(O)$ at $Q$ and intersects $BC$ at $P$. The circle with diameter $BP$ intersects $AB, BO$ at $K,S$ in this order. a. Prove that $SR$ passes through the midpoint of $KP$. b. Let $N$ be the midpoint of $BC$. The radical axis of circles with diameters $AN, BM$ intersects $SR$ at $E$. Prove that $ME$ always passes through a fixed point.

Given $2017$ positive real numbers $a_1,a_2,\dots ,a_{2017}$. For each $n>2017$, set $$a_n=\max\{ a_{i_1}a_{i_2}a_{i_3}|i_1+i_2+i_3=n, 1\leq i_1\leq i_2\leq i_3\leq n-1\}.$$Prove that there exists a positive integer $m\leq 2017$ and a positive integer $N>4m$ such that $a_na_{n-4m}=a_{n-2m}^2$ for every $n>N$.

For each integer $n>0$, a permutation $a_1,a_2,\dots ,a_{2n}$ of $1,2,\dots 2n$ is called beautiful if for every $1\leq i<j \leq 2n$, $a_i+a_{n+i}=2n+1$ and $a_i-a_{i+1}\not \equiv a_j-a_{j+1}$ (mod $2n+1$) (suppose that $a_{2n+1}=a_1$). a. For $n=6$, point out a beautiful permutation. b. Prove that there exists a beautiful permutation for every $n$.