Homework problems

Problem 1. One million trees grow in the forest. It is known that no pine tree has more than 600000 pine needles on it. Show that two pine trees in the forest must have the same number of pine needles.

Problem 2. In the country of Courland there are M soccer teams, each of which has 11players. All the players are gathered at an airport for a trip to another country for an important game, but they are traveling on ``standby''. There are 10 flights to their destination and it turns out that each flight has room for exactly M players. One soccer player will take his own helicopter to the game, rather that traveling standby on a plane. Show that at least one whole team will be sure to get to the important game.

Problem 3. What is the largest number of kings which could be placed on a chess board so that no two of them put each other in check?

Problem 4. Eleven students have formed five study groups in a summer camp.Prove that two students can be found, say A and B,such that every study group which includes student A includes B too.