Two circles, $\omega_1$ and $\omega_2$, centered at $O_1$ and $O_2$, respectively, meet at points $A$ and $B$. A line through $B$ meet $\omega_1$ again at $C$, and $\omega_2$ again at $D$. The tangents to $\omega_1$ and $\omega_2$ at $C$ and $D$, respectively, meet at $E$, and the line $AE$ meets the circle $\omega$ through $A, O_1,O_2$ again at $F$. Prove that the length of the segment $EF$ is equal to the diameter of $\omega$.

Let $n$ be a positive integer, and let $S_1,S_2,…,S_n$ be a collection of finite non-empty sets such that $$\sum_{1\leq i<j\leq n}{\frac{|S_i \cap S_j|}{|S_i||S_j|}} <1.$$Prove that there exist pairwise distinct elements $x_1,x_2,…,x_n$ such that $x_i$ is a member of $S_i$ for each index $i$.

Let $n$ be a positive integer, and let $a_1,a_2,..,a_n$ be pairwise distinct positive integers. Show that $$\sum_{k=1}^{n}{\frac{1}{[a_1,a_2,…,a_k]}} <4,$$where $[a_1,a_2,…,a_k]$ is the least common multiple of the integers $a_1,a_2,…,a_k$.

Determine the integers $k\geq 2$ for which the sequence $\Big\{ \binom{2n}{n} \pmod{k}\Big\}_{n\in \mathbb{Z}_{\geq 0}}$ is eventually periodic.

Given positive integers $k$ and $m$, show that $m$ and $\binom{n}{k}$ are coprime for infinitely many integers $n\geq k$.

Let $ABC$ be an acute triangle and let $M$ be the midpoint of $AC$. A circle $\omega$ passing through $B$ and $M$ meets the sides $AB$ and $BC$ at points $P$ and $Q$ respectively. Let $T$ be the point such that $BPTQ$ is a parallelogram. Suppose that $T$ lies on the circumcircle of $ABC$. Determine all possible values of $\frac{BT}{BM}$.

Prove that: (a) If $(a_n)_{n\geq 1}$ is a strictly increasing sequence of positive integers such that $\frac{a_{2n-1}+a_{2n}}{a_n}$ is a constant as $n$ runs through all positive integers, then this constant is an integer greater than or equal to $4$; and (b) Given an integer $N\geq 4$, there exists a strictly increasing sequene $(a_n)_{n\geq 1}$ of positive integers such that $\frac{a_{2n-1}+a_{2n}}{a_n}=N$ for all indices $n$.

Given any positive integer $n$, prove that: (a) Every $n$ points in the closed unit square $[0,1]\times [0,1]$ can be joined by a path of length less than $2\sqrt{n}+4$; and (b) There exist $n$ points in the closed unit square $[0,1]\times [0,1]$ that cannot be joined by a path of length less than $\sqrt{n}-1$.

Given a positive integer $n$, determine all functions $f$ from the first $n$ positive integers to the positive integers, satisfying the following two conditions: (1) $\sum_{k=1}^{n}{f(k)}=2n$; and (2) $\sum_{k\in K}{f(k)}=n$ for no subset $K$ of the first $n$ positive integers.

Given a positive integer $k$ and an integer $a\equiv 3 \pmod{8}$, show that $a^m+a+2$ is divisible by $2^k$ for some positive integer $m$.

Given a positive integer $n$, show that for no set of integers modulo $n$, whose size exceeds $1+\sqrt{n+4}$, is it possible that the pairwise sums of unordered pairs be all distinct.

Let $ABCD$ be a convex quadrilateral, and let $P$, $Q$, $R$, and $S$ be points on the sides $AB$, $BC$, $CD$, and $DA$, respectively. Let the line segment $PR$ and $QS$ meet at $O$. Suppose that each of the quadrilaterals $APOS$, $BQOP$, $CROQ$, and $DSOR$ has an incircle. Prove that the lines $AC$, $PQ$, and $RS$ are either concurrent or parallel to each other.

Determine the planar finite configurations $C$ consisting of at least $3$ points, satisfying the following conditions; if $x$ and $y$ are distinct points of $C$, there exist $z\in C$ such that $xyz$ are three vertices of equilateral triangles

Let $ABC$ be a triangle with $CA \neq CB$. Let $D$, $F$, and $G$ be the midpoints of the sides $AB$, $AC$, and $BC$ respectively. A circle $\Gamma$ passing through $C$ and tangent to $AB$ at $D$ meets the segments $AF$ and $BG$ at $H$ and $I$, respectively. The points $H'$ and $I'$ are symmetric to $H$ and $I$ about $F$ and $G$, respectively. The line $H'I'$ meets $CD$ and $FG$ at $Q$ and $M$, respectively. The line $CM$ meets $\Gamma$ again at $P$. Prove that $CQ = QP$. Proposed by El Salvador

Given a prime $p$, prove that the sum $\sum_{k=1}^{\lfloor \frac{q}{p} \rfloor}{k^{p-1}}$ is not divisible by $q$ for all but finitely many primes $q$.

Determine the positive integers expressible in the form $\frac{x^2+y}{xy+1}$, for at least $2$ pairs $(x,y)$ of positive integers

Determine all $f:\mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ such that $f(m)\geq m$ and $f(m+n) \mid f(m)+f(n)$ for all $m,n\in \mathbb{Z}^+$

A set $S=\{ s_1,s_2,...,s_k\}$ of positive real numbers is "polygonal" if $k\geq 3$ and there is a non-degenerate planar $k-$gon whose side lengths are exactly $s_1,s_2,...,s_k$; the set $S$ is multipolygonal if in every partition of $S$ into two subsets,each of which has at least three elements, exactly one of these two subsets in polygonal. Fix an integer $n\geq 7$. (a) Does there exist an $n-$element multipolygonal set, removal of whose maximal element leaves a multipolygonal set? (b) Is it possible that every $(n-1)-$element subset of an $n-$element set of positive real numbers be multipolygonal?