44th Eötvös Competition Problems 1940

1.  Each button in a box is big or small, and white or black. There is at least one big button, at least one small button, at least one white button, and at least one black button. Show that there are two buttons with different size and color. 2.  m and n are different positive integers. Show that 22m + 1 and 22n + 1 are coprime.