IZhO 2009

Find all pairs of integers $(x,y)$ , such that $$x^2 - 2009y + 2y^2 = 0$$

Find all real $a$ , such that there exist a function $f: \mathbb{R}\rightarrow\mathbb{R}$ satisfying the following inequality: $$x+af(y)\leq y+f(f(x))$$ for all $x,y\in\mathbb{R}$

For a convex hexagon $ABCDEF$ with an area $S$ , prove that: $$AC\cdot(BD+BF-DF)+CE\cdot(BD+DF-BF)+AE\cdot(BF+DF-BD)\geq 2\sqrt{3}S$$

On the plane, a Cartesian coordinate system is chosen. Given points $A_1,A_2,A_3,A_4$ on the parabola $y = x^2$ , and points $B_1,B_2,B_3,B_4$ on the parabola $y = 2009x^2$ . Points $A_1,A_2,A_3,A_4$ are concyclic, and points $A_i$ and $B_i$ have equal abscissas for each $i = 1,2,3,4$ . Prove that points $B_1,B_2,B_3,B_4$ are also concyclic.

Given a quadrilateral $ABCD$ with $\angle B=\angle D=90^{\circ}$ . Point $M$ is chosen on segment $AB$ so taht $AD=AM$ . Rays $DM$ and $CB$ intersect at point $N$ . Points $H$ and $K$ are feet of perpendiculars from points $D$ and $C$ to lines $AC$ and $AN$ , respectively. Prove that $\angle MHN=\angle MCK$ .

In a checked $17\times 17$ table, $n$ squares are colored in black. We call a line any of rows, columns, or any of two diagonals of the table. In one step, if at least $6$ of the squares in some line are black, then one can paint all the squares of this line in black. Find the minimal value of $n$ such that for some initial arrangement of $n$ black squares one can paint all squares of the table in black in some steps.