Problem 1: There is an unlimited pile of balls. They are numbered by 1, 2, 3....... Two players A and B randomly select one ball each and check their number. The winner is the player who picked the lowest numbered ball. Let us consider the chances of winning of player A. Assume that player A picked a ball with number k. As there is a finite number of balls numbered lower than k and unlimited number of ball with larger numbers, then we can conclude that the chance of player A winning is much more likely than loosing. However, the same argument works for the player B. Thus, both players have advantage over the other player. How is it possible?
Problem 2: Think of 10 cups with 1 marble in the first cup, 2 marbles in the second cup, 3 marbles in the third cup, and so on. Show that if cups are arranged in a circle in any order, then some three adjacent cups in the circle must contain a total of at least 17 marbles.