1.During a certain movie, each person of a set of five audience members fell asleep exactly twice. For each pair of these people, there was some moment when both were sleeping simultaneously. Is it true that, at some moment, three of them must have been sleeping simultaneously?
2. Given a set of two million points randomly placed on a piece of paper. Can you always draw a straight line on the paper so that on each side of the line there are exactly one million points?
3. A group of friends play a round-robin chess tournament, which means that everyone plays a game with everyone else exactly once. The tournament rules do not allow draws. Is it always possible to line up the players in such a way that the first player beats the second, who beats the third, etc.? Would this way of lining up the players be unique?
4. At the beginning of the morning the conference room in a large hall stands empty. Each minute, either one person enters or two people leave. The guestbook for the conference room shows that after exactly 31999 minutes, the room contained exactly 31000 + 2 people. Is the guestbook record accurate?
5. An alien lock has 16 keys arranged in a 4 × 4 grid, each key is either pointing horizontally or vertically. In order to open the lock, you must make sure that all the keys must be vertically oriented, by switching the orientation of one key at a time. When a key is switched to another position, all the other keys in the same row and column automatically switch their orientations too (i.e. vertical to horizontal, horizontal to vertical). Is it true that no matter what the initial positions of the 16 keys are, it is always possible to open this lock?
Please help me with these tricky questions! Thank you very much