2007 Polish Mathematical Olympiad Third Round
In an acute triangle let be the center of the circumcircle, segment be the height, point be any point on the segment and point be the middle of segment . The line perpendicular to and passing through passes the lines accordingly in points . Prove that .
We call a positive integer if it equals 1 or is the product of an even number of prime numbers (not necessarily distinct). All the other positive integers will be called . Examine if there exists any positive integer such that sum of its white divisors equals the sum of its black divisors.
The plane has been divided with horizontal and vertical lines into unit squares. In each square we write one positive integer so that every positive integer appears only once in the plane. Is it possible to write the numbers in such a way that every number is the divisor of the sum of numbers in 4 neighbour squares.?
Let be an integer. Determine the number of possible values of the product , where are integers satisfying the inequalities .
Tetrahedron satisfies the equalities Prove that the center of sphere circumscribed on lies on the line passing through the centres of edges and .
Sequence is described by conditions: and Prove that for .