2009 Polish Mathematical Olympiad Third Round

Day 1

Problem 1

Each of the vertices of a convex hexagon is a center of a circle of radius equal to the not longer side of the hexagon which includes that vertex. Prove that if the common part of all six circles (including the edge) is not empty, then the hexagon is regular.

Problem 2

Let $S$ be the set of all points of the plane with integer coordinates. Find the smallest positive integer $k$ such that there exists subset of $S$ with the following property: For any two different elements $A,B$ of that subset there exists a point $C \in S$ such that the area of triangle $ABC$ is equal to $k$ .

Problem 3

Let $P,Q,R$ be polynomials of degree at least one, with real coefficients, satisfying for all real numbers $x$ the equalities $P(Q(x)) = Q(R(x)) = R(P(x)).$ Prove that $P=Q=R$ .

Day 2

Problem 4

Let $x_1, x_2, \ldots, x_n$ be nonnegative numbers with sum equal to 1. Prove that there exist numbers $a_1, a_2, \ldots, a_n \in \{0,1,2,3,4\}$ such that $(a_1, a_2, \ldots, a_n) \neq (2,2,\ldots,2)$ and $2 \leq a_1x_1 + a_2x_2 + \ldots + a_nx_n \leq 2 + \frac{2}{3^n - 1}$ .

Problem 5

Sphere inscribed in tetrahedron $ABCD$ is tangent to its sides $BCD,ACD,ABD,ABC$ respectively in points $P,Q,R,S$ . The segment $PT$ is the diameter of that sphere and points $A',Q',R',S'$ are the intersections of lines $TA,TQ,TR,TS$ with the plane $BCD$ . Prove that $A'$ is the center of the circumcircle of triangle $Q'R'S'$ .

Problem 6

Let $n \geq 3$ be a natural number. The sequence of nonnegative numbers $(c_0,c_1, \ldots, c_n)$ satisfies the condition $c_pc_s + c_rc_t = c_{p+r}c_{r+s}$ for all $p,r,s,t \geq 0$ such that $p+r+s+t = n$ . Find all possible values of $c_2$ if $c_1=1$ .

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