HMMT-Nov 2010 thm, Problem 1

16 progamers are playing in a single elimination tournament. Each player has a different skill level and when two play against each other the one with the higher skill level will always win. Each round, each progamer plays a match against another and the loser is eliminated. This continues until only one remains. How many different progamers can reach the round that has 2 players remaining?

Solution: Each finalist must be better than the person he beat in the semifinals, both of the people they beat in the second round, and all 4 of the people any of those people beat in the first round. So, none of the 7 worst players can possibly make it to the finals. Any of the 9 best players can make it to the finals if the other 8 of the best 9 play each other in all rounds before the finals. So, exactly 9 people are capable of making it to the finals. $\fbox{9}$.

Answer: 9