There is a board with a number written in each cell. Every two neighbour rows sum up to at least , and every two neighbour columns sum up to at most . Find the sum of all numbers on the board.

A square is divided into unit squares. Is it possible to fill each unit square with a number in such a way that, whenever one places the tile so that it fully covers nine unit squares, the tile will cover nine different numbers?

A quadrilateral with no parallel sides is inscribed in a circle . Circles are inscribed in triangles , respectively. Common external tangents are drawn between and , and , and , and and , not containing any sides of quadrilateral . A quadrilateral whose consecutive sides lie on these four lines is inscribed in a circle . Prove that the lines joining the centers of and , and , and the centers of and all intersect at one point.

We denote the number of positive divisors of a positive integer by and the number of distinct prime divisors of by . Let be a positive integer. Prove that there exist infinitely many positive integers such that and does not divide for any positive integers satisfying .

Find all functions such that for any :

Solve the system of equations

Find all functions such that for all

Let be a triangle with circumcircle and let be the -excircle. Let and be the intersection points of and . Let and be the projections of onto the tangent lines to at and respectively. The tangent line at to the circumcircle of the triangle intersects the tangent line at to the circumcircle of the triangle at a point . Prove that .

Let and : Prove that :

The school A has m1 boys and m2 girls, and ,the school B has n1 boys and n2 girls. Each school is represented by one team formed by p students,boys and girls. If f(k) is the number of cases for which,the twice schools has,togheter k girls, fund f(k) and the valute of k, for which f(k) is maximum.