Positive integers $a,b,c$ satisfy the equations $a^2=b^3+ab$ and $c^3=a+b+c$. Prove that $a=bc$.

Given an integer $n>1$. The board $n\times n$ is colored white and black in a chess-like manner. We call any non-empty set of different cells of the board as a figure. We call figures $F_1$ and $F_2$ similar, if $F_1$ can be obtained from $F_2$ by a rotation with respect to the center of the board by an angle multiple of $90^\circ$ and a parallel transfer. (Any figure is similar to itself.) We call a figure $F$ connected if for any cells $a,b\in F$ there is a sequence of cells $c_1,\ldots,c_m\in F$ such that $c_1 = a$, $c_m = b$, and also $c_i$ and $c_{i+1}$ have a common side for each $1\le i\le m - 1$. Find the largest possible value of $k$ such that for any connected figure $F$ consisting of $k$ cells, there are figures $F_1,F_2$ similar to $F$ such that $F_1$ has more white cells than black cells and $F_2$ has more black cells than white cells in it.

An acute triangle $ABC$ ($AB\ne AC$) is inscribed in the circle $\omega$ with center at $O$. The point $M$ is the midpoint of the side $BC$. The tangent line to $\omega$ at point $A$ intersects the line $BC$ at point $D$. A circle with center at point $M$ with radius $MA$ intersects the extensions of sides $AB$ and $AC$ at points $K$ and $L$, respectively. Let $X$ be such a point that $BX\parallel KM$ and $CX\parallel LM$. Prove that the points $X$, $D$, $O$ are collinear.

Prove that for any positive integers $a$, $b$, $c$, at least one of the numbers $a^3b+1$, $b^3c+1$, $c^3a+1$ is not divisible by $a^2+b^2+c^2$.

In triangle $ABC$ ($AB\ne AC$), where all angles are greater than $45^\circ$, the altitude $AD$ is drawn. Let $\omega_1$ and $\omega_2$ be-- circles with diameters $AC$ and $AB$, respectively. The angle bisector of $\angle ADB$ secondarily intersects $\omega_1$ at point $P$, and the angle bisector of $\angle ADC$ secondarily intersects $\omega_2$ at point $Q$. The line $AP$ intersects $\omega_2$ at the point $R$. Prove that the circumcenter of triangle $PQR$ lies on line $BC$.

An integer $m\ge 3$ and an infinite sequence of positive integers $(a_n)_{n\ge 1}$ satisfies the equation \[a_{n+2} = 2\sqrt[m]{a_{n+1}^{m-1} + a_n^{m-1}} - a_{n+1}. \]for all $n\ge 1$. Prove that $a_1 < 2^m$.

Let $ABC$ be an acute triangle with an altitude $AD$. Let $H$ be the orthocenter of triangle $ABC$. The circle $\Omega$ passes through the points $A$ and $B$, and touches the line $AC$. Let $BE$ be the diameter of $\Omega$. The lines~$BH$ and $AH$ intersect $\Omega$ for the second time at points $K$ and $L$, respectively. The lines $EK$ and $AB$ intersect at the point~$T$. Prove that $\angle BDK=\angle BLT$.

Given a prime number $p\ge 3,$ and an integer $d \ge 1$. Prove that there exists an integer $n\ge 1,$ such that $\gcd(n,d) = 1,$ and the product \[P=\prod\limits_{1 \le i < j < p} {({i^{n + j}} - {j^{n + i}})} \text{ is not divisible by } p^n.\]

Find all functions $f: \mathbb R^+ \to \mathbb R^+$ such that \[ f \left( x+\frac{f(xy)}{x} \right) = f(xy) f \left( y + \frac 1y \right) \]holds for all $x,y\in\mathbb R^+.$

Players $A$ and $B$ play the following game on the coordinate plane. Player $A$ hides a nut at one of the points with integer coordinates, and player $B$ tries to find this hidden nut. In one move $B$ can choose three different points with integer coordinates, then $A$ tells whether these three points together with the nut's point lie on the same circle or not. Can $B$ be guaranteed to find the nut in a finite number of moves?

The circle $\omega$ with center at point $I$ inscribed in an triangle $ABC$ ($AB\neq AC$) touches the sides $BC$, $CA$ and $AB$ at points $D$, $E$ and $F$, respectively. The circumcircles of triangles $ABC$ and $AEF$ intersect secondary at point $K.$ The lines $EF$ and $AK$ intersect at point $X$ and intersects the line $BC$ at points $Y$ and $Z$, respectively. The tangent lines to $\omega$, other than $BC$, passing through points $Y$ and $Z$ touch $\omega$ at points $P$ and $Q$, respectively. Let the lines $AP$ and $KQ$ intersect at the point $R$. Prove that if $M$ is a midpoint of segment $YZ,$ then $IR\perp XM$.