International Competitions

Let $x, y, z$ be real numbers satisfying $x^2 + y^2 + z^2 + 9 = 4(x + y + z)$. Prove that $$x^4 + y^4 + z^4 + 16(x^2 + y^2 + z^2) \geqslant 8(x^3 + y^3 + z^3) + 27$$ and determine when equality holds.

Let $a, b, c$ be real numbers such that for every two of the equations $$x^2 + ax + b = 0, \quad x^2 + bx + c = 0, \quad x^2 + cx + a = 0$$ there is exactly one real number satisfying both of them. Determine all the possible values of $a^2 + b^2 + c^2$.

The numbers $0, 1, 2, \ldots, n$ ($n \geqslant 2$) are written on a blackboard. In each step we erase an integer which is the arithmetic mean of two different numbers which are still left on the blackboard. We make such steps until no further integer can be erased. Let $g(n)$ be the smallest possible number of integers left on the blackboard at the end. Find $g(n)$ for every $n$.

We colour every square of the $2009 \times 2009$ board with one of $n$ colours (we do not have to use every colour). A colour is called connected if either there is only one square of that colour or any two squares of the colour can be reached from one another by a sequence of moves of a chess queen without intermediate stops at squares having another colour (a chess queen moves horizontally, vertically or diagonally). Find the maximum $n$, such that for every colouring of the board at least one colour present at the board is connected.

Let $ABCD$ be a parallelogram with $\measuredangle BAD = 60°$ and denote by $E$ the intersection of its diagonals. The circumcircle of the triangle $ACD$ meets the line $BA$ at $K \neq A$, the line $BD$ at $P \neq D$ and the line $BC$ at $L \neq C$. The line $EP$ intersects the circumcircle of the triangle $CEL$ at points $E$ and $M$. Prove that the triangles $KLM$ and $CAP$ are congruent.

Suppose that $ABCD$ is a cyclic quadrilateral and $CD = DA$. Points $E$ and $F$ belong to the segments $AB$ and $BC$ respectively, and $\measuredangle ADC = 2\measuredangle EDF$. Segments $DK$ and $DM$ are height and median of the triangle $DEF$, respectively. $L$ is the point symmetric to $K$ with respect to $M$. Prove that the lines $DM$ and $BL$ are parallel.

Find all pairs $(m, n)$ of integers which satisfy the equation $$(m + n)^4 = m^2n^2 + m^2 + n^2 + 6mn.$$

Find all non-negative integer solutions of the equation $$2^x + 2009 = 3^y5^z.$$