TJUSAMO

Wow, TJUSAMO was huge yesterday. There must have been at least 15 people, and another 20 in TJAIME. Not even the first practices have been this big in past years. Hopefully everyone will keep coming. That would be pretty sweet. I decided for this year that since David Yang got a 27 on USAMO with horrible proof writing skills that I didn't need to teach people what a proof is, and they can pick it up on the fly. So I gave them the following four problems. Maybe you guys will enjoy them as well:

1. Consider an analog clock with an hour hand and a minute hand that move continuously.
   a) How many times (in a 12-hour period) are there such that when you switch the hour and the minute hand they still form a valid time?
   b) What is the first such time after 1:00?
2. You have n cubes of sizes 1, 2, 3, ..., n. You want to build a tower out of these cubes, but in this tower the cube on top of the cube of size k must be at most size k+2 for every k. How many ways to build this tower are there?
3. a) There are six people in a room. For each pair, they are either friends or enemies. Show that there are three of them such that each is either friends with the other two or each is enemies with the other two.
   b) There is another room where each pair of people is either friends, enemies, or they don't know each other. How many people do there need to be to guarantee that there are three people such that each is friends with the other two, each is enemies with the other two, or no two of them know each other?
4. The numbers 2, 3, ..., 2010 are written on the board. Then someone comes up with the idea that they should repeatedly take two numbers x and y on the board and replace them by the number (x+y)/(1+xy) until only one number remains. What are all possibilities for this last number?