2009 Zhautykov International Olympiad Problems

Day 1

Problem 1

Find all pairs of integers $(x,y)$ , such that

$x^2 - 2009y + 2y^2 = 0$ .

Problem 2

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 $x,y\in\mathbb{R}$ .

Problem 3

For a convex hexagon $ABCDEF$ with an area $S$ , prove that:

$AC\cdot(BD + BF - DF) + CE\cdot(BD + DF - BF) + AE\cdot(BF + DF - BD)\geq 2\sqrt {3}$ .

Day 2

Problem 4

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.

Problem 5

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 $\angle MHN = \angle MCK$ .

Problem 6

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.

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