may's problem

The Handshakes

Every person on earth has shaken a certain number of hands. Prove that the number of persons who have shaken an odd number of hands is even.

April's problem

The Lucky Prisoners

A jailer, carrying out the terms of a partial amnesty, unlocked every cell in the prison row. Next he locked every second cell. Then he turned the key in every third cell, locking those cells which were open and opening those which were locked. He continued this way, on the n^th trip turning the key in every n^th cell. Those prisoners whose cells eventually remained open were allowed to go free. Who were the lucky ones?

Answer