In a tennis tournament, each player advances to the next round only in case of victory. If it is not possible for an even number of players to always advance to the next round, the tournament organizers decide which rounds certain players should play. For example, a seeded player can, at the discretion of the organizers, enter the second round, or advance from the first to the third round, so that the total number of players participating in each round is even.

(a) Consider a tennis tournament with 64 players. How many matches are played?

(b) And in a tournament with 2011 players?

##
(a) In the first round, 32 matches are played, from which 32 players advance to the next phase. Then 16 matches are played, qualifying 16 for the next round, and so on. Thus, the number of matches in the tournament is

$$
32+16+8+4+2+1=63
$$

(b) Since in each match one player is eliminated, the number of matches is equal to the number of players eliminated, that is, $2011-1=2010$.

## Related Problem

A football tournament with 57 teams will be played according to the following rules:

(a) No game can end in a tie.

(b) The team that loses two matches will be eliminated.

(c) The tournament ends when only one team remains, which will be the champion.

If the champion team loses once, how many matches will be played in the tournament?