AMC 10 Weekly Practice Round 3
Complete problem set with solutions and individual problem pages
There are players in a chess qualifying tournament. Each pair of players plays exactly one game. In each game, the winner gets point and the loser gets points; if the game is a draw, each player gets points. After all the games are finished, players with at least points can advance. What is the maximum possible number of players who can advance in this tournament?
- A.
- B.
- C.
- D.
- E.
Sixteen players play a total of games, and the total number of points is . Since a player must score at least points to qualify, the number of qualifiers cannot exceed
First, we prove that having qualifiers is impossible. Suppose there are qualifiers; then there would be non-qualifiers. The non-qualifiers play games among themselves, earning a total of points. Thus, the qualifiers can earn at most points in total. Since the sum of the players’ points is , at least one of them must have fewer than points, contradicting the assumption that all are qualifiers. Hence, there can be at most qualifiers.
Next, we show that having qualifiers is possible. Among the players, there are games, totaling points. If all these games end in draws, each player gets points. In addition, each of these players must play against the other non-qualifiers; if each of these games is won, then each player earns another points. Therefore, each of the players has points, meeting the qualifying standard.
Thus, the answer is .
