An acute triangle $ABC$ is given. Let $D$ be the foot of altitude dropped for $A$. Tangents from $D$ to circles with diameters $AB$ and $AC$ intersects with the said circles at $K$ and $L$, in respective. Point $S$ in the plane is given so that $\angle ABC + \angle ABS = \angle ACB + \angle ACS = 180^\circ$. Prove that $A, K, L$ and $S$ lie on a circle.

Given an acute triangle $ABC$ let $M$ be the midpoint of $AB$. Point $K$ is given on the other side of line $AC$ from that of point $B$ such that $\angle KMC = 90 ^ \circ $ and $\angle KAC = 180^\circ - \angle ABC$. The tangent to circumcircle of triangle $ABC$ at $A$ intersects line $CK$ at $E$. Prove that the reflection of line $BC$ with respect to $CM$ passes through the midpoint of line segment $ME$.

Given triangle $ABC$ variable points $X$ and $Y$ are chosen on segments $AB$ and $AC$, respectively. Point $Z$ on line $BC$ is chosen such that $ZX=ZY$. The circumcircle of $XYZ$ cuts the line $BC$ for the second time at $T$. Point $P$ is given on line $XY$ such that $\angle PTZ = 90^ \circ$. Point $Q$ is on the same side of line $XY$ with $A$ furthermore $\angle QXY = \angle ACP$ and $\angle QYX = \angle ABP$. Prove that the circumcircle of triangle $QXY$ passes through a fixed point (as $X$ and $Y$ vary).

Let $S$ be an infinite set of positive integers, such that there exist four pairwise distinct $a,b,c,d \in S$ with $\gcd(a,b) \neq \gcd(c,d)$. Prove that there exist three pairwise distinct $x,y,z \in S$ such that $\gcd(x,y)=\gcd(y,z) \neq \gcd(z,x)$.

Is it possible to arrange a permutation of Integers on the integer lattice infinite from both sides such that each row is increasing from left to right and each column increasing from up to bottom?

Let $n\ge 3$ be a fixed integer. There are $m\ge n+1$ beads on a circular necklace. You wish to paint the beads using $n$ colors, such that among any $n+1$ consecutive beads every color appears at least once. Find the largest value of $m$ for which this task is $\emph{not}$ possible. Carl Schildkraut, USA

Positive real numbers $a, b, c$ and $d$ are given such that $a+b+c+d = 4$ prove that $$\frac{ab}{a^2-\frac{4}{3}a+\frac{4}{3}} + \frac{bc}{b^2-\frac{4}{3}b+ \frac{4}{3}} + \frac{cd}{c^2-\frac{4}{3}c+ \frac{4}{3}} + \frac{da}{d^2-\frac{4}{3}d+ \frac{4}{3}}\leq 4.$$

If $a, b, c$ and $d$ are complex non-zero numbers such that $$2|a-b|\leq |b|, 2|b-c|\leq |c|, 2|c-d| \leq |d| , 2|d-a|\leq |a|.$$Prove that $$\frac{7}{2} <\left| \frac{b}{a} + \frac{c}{b} + \frac{d}{c} + \frac{a}{d} \right| .$$

Polynomial $P$ with non-negative real coefficients and function $f:\mathbb{R}^+\to \mathbb{R}^+$ are given such that for all $x, y\in \mathbb{R}^+$ we have $$f(x+P(x)f(y)) = (y+1)f(x)$$(a) Prove that $P$ has degree at most 1. (b) Find all function $f$ and non-constant polynomials $P$ satisfying the equality.

For a natural number $n$, $f(n)$ is defined as the number of positive integers less than $n$ which are neither coprime to $n$ nor a divisor of it. Prove that for each positive integer $k$ there exist only finitely many $n$ satisfying $f(n) = k$.

Find all functions $f:\mathbb{N} \to \mathbb{N}$ such that for any two positive integers $a$ and $b$ we have $$ f^a(b) + f^b(a) \mid 2(f(ab) +b^2 -1)$$Where $f^n(m)$ is defined in the standard iterative manner.

$x_1$ is a natural constant. Prove that there does not exist any natural number $m> 2500$ such that the recursive sequence $\{x_i\} _{i=1} ^ \infty $ defined by $x_{n+1} = x_n^{s(n)} + 1$ becomes eventually periodic modulo $m$. (That is there does not exist natural numbers $N$ and $T$ such that for each $n\geq N$, $m\mid x_n - x_{n+T}$). ($s(n)$ is the sum of digits of $n$.)

Is it possible to arrange natural numbers 1 to 8 on vertices of a cube such that each number divides sum of the three numbers sharing an edge with it?

Given an acute triangle $ABC$, let $AD$ be an altitude and $H$ the orthocenter. Let $E$ denote the reflection of $H$ with respect to $A$. Point $X$ is chosen on the circumcircle of triangle $BDE$ such that $AC\| DX$ and point $Y$ is chosen on the circumcircle of triangle $CDE$ such that $DY\| AB$. Prove that the circumcircle of triangle $AXY$ is tangent to that of $ABC$.

Find all functions $f: \mathbb{Q}[x] \to \mathbb{R}$ such that: (a) for all $P, Q \in \mathbb{Q}[x]$, $f(P \circ Q) = f(Q \circ P);$ (b) for all $P, Q \in \mathbb{Q}[x]$ with $PQ \neq 0$, $f(P\cdot Q) = f(P) + f(Q).$ ($P \circ Q$ indicates $P(Q(x))$.)

Arash and Babak play the following game, taking turns alternatively, on a $1400\times 1401$ table. Arash starts and in his turns he colors $k$, $L$-corners (any three cell of a square). Babak in his turn colors one $2\times 2$ square. Neither player is allowed to recolor any cell. Find all positive integers $k$ for which Arash has a winning strategy.