AMC10 2003 B

AMC10 2003 B · Q15

AMC10 2003 B · Q15. It mainly tests Basic counting (rules of product/sum), Casework.

There are 100 players in a singles tennis tournament. The tournament is single elimination, meaning that a player who loses a match is eliminated. In the first round, the strongest 28 players are given a bye, and the remaining 72 players are paired off to play. After each round, the remaining players play in the next round. The match continues until only one player remains unbeaten. The total number of matches played is

单打网球锦标赛有 100 名选手。采用单淘汰制，输掉比赛的选手被淘汰。第一轮，最强的 28 名选手轮空，其余 72 名选手两两配对比赛。每轮后，剩余选手进入下一轮。比赛继续直到仅剩一名不败选手。总共进行了多少场比赛？

(A) a prime number 素数

(B) divisible by 2 能被 2 整除

(D) divisible by 7 能被 7 整除

(E) divisible by 11 能被 11 整除

Answer

Correct choice: (E)

正确答案：（E）

Solution

(E) In the first round $100-64=36$ players are eliminated, one per match. In the second round there are 32 matches, in the third 16, then 8, 4, 2, and 1. The total number of matches is: $36+32+16+8+4+2+1=99$. Note that 99 is divisible by 11, but 99 does not satisfy any of the other conditions given as answer choices.

（E）第一轮中有 $100-64=36$ 名选手被淘汰，每场比赛淘汰一人。第二轮有 32 场比赛，第三轮 16 场，然后分别是 8、4、2 和 1 场。比赛总场数为： $36+32+16+8+4+2+1=99$。注意，99 能被 11 整除，但 99 不满足答案选项中给出的其他任何条件。

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