KZ-MO 2006

Natural numbers from $1$ to $200$ were divided into $50$ sets. Prove that one of them contains three numbers that are the lengths of the sides of some triangle

Product of square trinomials $x ^ 2 + a_1x + b_1$ , $x ^ 2 + a_2x + b_2$ , $\dots$ , $x ^ 2 + a_n x + b_n$ equals polynomial $P (x) = x ^ {2n} + c_1x ^ {2n-1} + c_2x ^ {2n-2} + \dots + c_ {2n-1} x + c_ {2n}$ , where the coefficients $c_1$ , $c_2$ , $\dots$ , $c_ {2n}$ are positive. Prove that for some $k$ ( $1 \leq k \leq n$ ) the coefficients $a_k$ and $b_k$ are positive.

The racing tournament has $12$ stages and $n$ participants. After each stage, all participants, depending on the occupied place $k$ , receive points $a_k$ (the numbers $a_k$ are natural and $a_1> a_2> \dots> a_n$ ). For what is the smallest $n$ the tournament organizer can choose the numbers $a_1$ , $\dots$ , $a_n$ so that after the penultimate stage for any possible distribution of places at least two participants had a chance to take first place.

grade IX P4, X P3 The bisectors of the angles $A$ and $C$ of the triangle $ABC$ intersect the circumscirbed circle of this triangle at the points $A_0$ and $C_0$ , respectively. The straight line passing through the center of the inscribed circle of triangle $ABC$ parallel to the side of $AC$ , intersects with the line $A_0C_0$ at $P$ . Prove that the line $PB$ is tangent to the circumcircle of the triangle $ABC$ .

grade XI P4 The bisectors of the angles $A$ and $C$ of the triangle $ABC$ intersect the sides at the points $A_1$ and $C_1$ , and the circumcircle of this triangle at points $A_0$ and $C_0$ respectively. Straight lines $A_1C_1$ and $A_0C_0$ intersect at point $P$ . Prove that the segment connecting $P$ with the center inscribed circles of triangle $ABC$ , parallel to $AC$ .

Prove that for every $x$ such that $\sin x \neq 0$ , exists natural $n$ such that $| \sin nx | \geq \frac {\sqrt {3}} {2}$ .

In the tetrahedron $ABCD$ from the vertex $A$ , the perpendiculars $AB '$ , $AC'$ are drawn, $AD '$ on planes dividing dihedral angles at edges $CD$ , $BD$ , $BC$ in half. Prove that the plane $(B'C'D ')$ is parallel to the plane $(BCD)$ .

Prove that if a natural number $N$ can be represented in the form the sum of three squares of integers divisible by $3$ , then it is also is represented as the sum of three squares of integers that are not divisible by $3$ .

What is the minimum number of cells that can be colored black in white square $300 \times 300$ so that no three black cells formed a corner, and after painting any white cell this condition violated?