1. Prove that for any two positive integers there exist positive integers such that
2. Given red and blue points in the plane, no three of them on a line. We aim to split the plane by lines (not passing through these points) into regions such that there are no regions containing points of both the colors. What is the least number of lines that always suffice?
3. Let be a triangle and let , , and be points of contact of the excircles with the sides , , and , respectively. Prove that if the circumcenter of lies on the circumcircle of , then is a right triangle.