2009 Polish Mathematical Olympiad Third Round
Contents
[hide]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 be the set of all points of the plane with integer coordinates. Find the smallest positive integer such that there exists subset of with the following property: For any two different elements of that subset there exists a point such that the area of triangle is equal to .
Problem 3
Let be polynomials of degree at least one, with real coefficients, satisfying for all real numbers the equalities Prove that .
Day 2
Problem 4
Let be nonnegative numbers with sum equal to 1. Prove that there exist numbers such that and .
Problem 5
Sphere inscribed in tetrahedron is tangent to its sides respectively in points . The segment is the diameter of that sphere and points are the intersections of lines with the plane . Prove that is the center of the circumcircle of triangle .
Problem 6
Let be a natural number. The sequence of nonnegative numbers satisfies the condition for all such that . Find all possible values of if .