There are 44 distinct holes along a straight groove and 2017 ants. Each ant emerges from one hole, crawls at constant velocity to a different hole, and disappears into it. Let $T$ be the set of all emergence and disappearance times. The ants' velocities are pairwise different, and $|T|\le45$. Prove that at least two ants never meet. Two ants meet if, at some instant, both occupy the same position in the groove, including their emergence and disappearance instants.
For each positive integer $n$, let $x_n=\binom{2n}{n}$.
(a) Prove that if $2017^k/2\lt n\lt 2017^k$ for some positive integer $k$, then $2017\mid x_n$.
(b) Find every integer $h>1$ for which there exist positive integers $N,T$ such that $x_{n+T}\equiv x_n\pmod h$ for all $n>N$.
Triangle $ABC$ has incircle $(I)$ tangent to $BC,CA,AB$ at $D,E,F$, respectively. Let $I_b,I_c$ be its $B$- and $C$-excenters. Let $P,Q$ be the midpoints of $I_bE,I_cF$. The circles $(PAC),(QAB)$ meet $AB,AC$ again at $R,S$, respectively.
(a) Prove that $PR,QS,AI$ are concurrent.
(b) Put $K=DE\cap I_bI_c$, $J=DF\cap I_bI_c$, and $M=EJ\cap FK$. The lines $PE,QF$ meet $(PAC),(QAB)$ again at $X,Y$, respectively. Prove that $BY,CX,AM$ are concurrent.
Triangle $ABC$ is inscribed in a fixed circle $(O)$, with $B,C$ fixed and $A$ moving on $(O)$ so that $AB>BC$. Let $M$ be the midpoint of $AC$. The circle with diameter $BM$ meets $(O)$ again at $R$. The line $RM$ meets $(O)$ again at $Q$ and meets $BC$ at $P$. The circle with diameter $BP$ meets $AB$ and $BO$ again at $K$ and $S$, respectively.
(a) Prove that $SR$ passes through the midpoint of $KP$.
(b) Let $N$ be the midpoint of $BC$. The radical axis of the circles with diameters $AN$ and $BM$ meets $SR$ at $E$. Prove that $ME$ always passes through a fixed point.
Given 2017 positive real numbers $a_1,a_2,\ldots,a_{2017}$, define, for every $n>2017$,
$$a_n=\max\{a_{i_1}a_{i_2}a_{i_3}:i_1+i_2+i_3=n,\ 1\le i_1\le i_2\le i_3\le n-1\}.$$
Prove that there exist positive integers $m\le2017$ and $N>4m$ such that $a_na_{n-4m}=a_{n-2m}^2$ for all $n>N$.
For each positive integer $n$, consider a permutation $a_1,a_2,\ldots,a_{2n}$ of the first $2n$ positive integers. Call it beautiful if opposite entries satisfy $a_i+a_{n+i}=2n+1$ for $1\le i\le n$, and the $2n$ differences $a_i-a_{i+1}$ are pairwise incongruent modulo $2n+1$, with $a_{2n+1}=a_1$.
(a) Give a beautiful permutation for $n=6$.
(b) Prove that a beautiful permutation exists for every positive integer $n$.
