2009 Zhautykov International Olympiad Problems
Contents
[hide]Day 1
Problem 1
Find all pairs of integers , such that
Problem 2
Find all real , such that there exist a function satisfying the following inequality:
for all .
Problem 3
For a convex hexagon with an area , prove that:
Day 2
Problem 4
On the plane, a Cartesian coordinate system is chosen. Given points on the parabola , and points on the parabola . Points are concyclic, and points and have equal abscissas for each . Prove that points are also concyclic.
Problem 5
Given a quadrilateral with . Point is chosen on segment so taht . Rays and intersect at point . Points and are feet of perpendiculars from points and to lines and , respectively. Prove that .
Problem 6
In a checked table, 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 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 such that for some initial arrangement of black squares one can paint all squares of the table in black in some steps.