Let $a$, $b$, $c$ be real numbers, not all of them are equal. Prove that $a+b+c=0$ if and only if $a^2+ab+b^2=b^2+bc+c^2=c^2+ca+a^2$.

On a chessboard $8\times 8$, $n>6$ Knights are placed so that for any 6 Knights there are two Knights that attack each other. Find the greatest possible value of $n$.

Suppose that $a,b,c$ are positive integers such that $a^b$ divides $b^c$, and $a^c$ divides $c^b$. Prove that $a^2$ divides $bc$.

By centroid of a quadrilateral $PQRS$ we call a common point of two lines through the midpoints of its opposite sides. Suppose that $ABCDEF$ is a hexagon inscribed into the circle $\Omega$ centered at $O$. Let $AB=DE$, and $BC=EF$. Let $X$, $Y$, and $Z$ be centroids of $ABDE$, $BCEF$; and $CDFA$, respectively. Prove that $O$ is the orthocenter of triangle $XYZ$.

Baron Munсhausen discovered the following theorem: "For any positive integers $a$ and $b$ there exists a positive integer $n$ such that $an$ is a perfect square, while $bn$ is a perfect cube". Determine if the statement of Baron’s theorem is correct.

Given a convex quadrilateral $ABCD$ with $\angle BCD=90^\circ$. Let $E$ be the midpoint of $AB$. Prove that $2EC \leqslant AD+BD$.

Given a positive integer $n>1$. In the cells of an $n\times n$ board, marbles are placed one by one. Initially there are no marbles on the board. A marble could be placed in a free cell neighboring (by side) with at least two cells which are still free. Find the greatest possible number of marbles that could be placed on the board according to these rules.

Let $a, b, c$ be the lengths of sides of a triangle. Prove the inequality $$(a+b)\sqrt{ab}+(a+c)\sqrt{ac}+(b+c)\sqrt{bc} \geq (a+b+c)^2/2.$$

A tetrahedron is given. Determine whether it is possible to put some 10 consecutive positive integers at 4 vertices and at 6 midpoints of the edges so that the number at the midpoint of each edge is equal to the arithmetic mean of two numbers at the endpoints of this edge.

Let $I$ be the incenter of an acute-angled triangle $ABC$. Let $P$, $Q$, $R$ be points on sides $AB$, $BC$, $CA$ respectively, such that $AP=AR$, $BP=BQ$ and $\angle PIQ = \angle BAC$. Prove that $QR \perp AC$.

For $2n$ positive integers a matching (i.e. dividing them into $n$ pairs) is called {\it non-square} if the product of two numbers in each pair is not a perfect square. Prove that if there is a non-square matching, then there are at least $n!$ non-square matchings. (By $n!$ denote the product $1\cdot 2\cdot 3\cdot \ldots \cdot n$.)

Morteza places a function $[0,1]\to [0,1]$ (that is a function with domain [0,1] and values from [0,1]) in each cell of an $n \times n$ board. Pavel wants to place a function $[0,1]\to [0,1]$ to the left of each row and below each column (i.e. to place $2n$ functions in total) so that the following condition holds for any cell in this board: If $h$ is the function in this cell, $f$ is the function below its column, and $g$ is the function to the left of its row, then $h(x) = f(g(x))$ for all $x \in [0, 1]$. Prove that Pavel can always fulfil his plan.

Baron Munсhausen discovered the following theorem: "For any positive integers $a$ and $b$ there exists a positive integer $n$ such that $an$ is a perfect cube, while $bn$ is a perfect fifth power". Determine if the statement of Baron’s theorem is correct.

Two graphs $G_1$ and $G_2$ of quadratic polynomials intersect at points $A$ and $B$. Let $O$ be the vertex of $G_1$. Lines $OA$ and $OB$ intersect $G_2$ again at points $C$ and $D$. Prove that $CD$ is parallel to the $x$-axis.

In an acute-angled triangle $ABC$, the altitudes from $A,B,C$ meet the sides of $ABC$ at $A_1$, $B_1$, $C_1$, and meet the circumcircle of $ABC$ at $A_2$, $B_2$, $C_2$, respectively. Line $A_1 C_1$ intersects the circumcircles of triangles $AC_1 C_2$ and $CA_1 A_2$ at points $P$ and $Q$ ($Q\neq A_1$, $P\neq C_1$). Prove that the circle $PQB_1$ touches the line $AC$.

In the cells of an $8\times 8$ board, marbles are placed one by one. Initially there are no marbles on the board. A marble could be placed in a free cell neighboring (by side) with at least three cells which are still free. Find the greatest possible number of marbles that could be placed on the board according to these rules.