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2003 AMC 10B Problem 15

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2003 AMC 10B Problem 15AMC 10 Step by Step
Name:                             Date: September 13, 2026
Games & Processes · difficulty 2 of 5 · about 2 min · position P11-15

There are 100100 players in a single tennis tournament. The tournament is single elimination, meaning that a player who loses a match is eliminated. In the first round, the strongest 2828 players are given a bye, and the remaining 7272 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

$\qquad

  1. A)

    a prime number\text{a prime number}

  2. B)

    divisible by 2\text{divisible by 2}

  3. C)

    divisible by 5\text{divisible by 5}

  4. D)

    divisible by 7\text{divisible by 7}

  5. E)

    divisible by 11\text{divisible by 11}

Answer

E

Key insight

Every match eliminates exactly one player and 99 players must be eliminated, so there are 99 matches regardless of byes; 99 = 9 * 11.

Solution

Each match has exactly one loser, and a loser is eliminated and never plays again. The tournament ends when a single unbeaten player remains, so exactly 100−1=99100 - 1 = 99 players are eliminated over the whole tournament.

Matching each match to its loser is a one-to-one correspondence, so the number of matches equals the number of eliminated players: 9999. The byes and the pairings change when players are eliminated, not how many.

Since 99=9⋅1199 = 9 \cdot 11, the number of matches is divisible by 1111 (and it is odd, not prime, not divisible by 55 or 77).

The answer is (E) divisible by 11\boxed{\textbf{(E)}\ \text{divisible by 11}}.

Why this works

In any single-elimination event with nn entrants, the number of matches is n−1n - 1, because matches and eliminations are in bijection. The bye structure is pure distraction. When a process "removes one object per step," count the objects removed instead of simulating the steps.

Alternative approach

Simulate: round one has 72/2=3672/2 = 36 matches, leaving 36+28=6436 + 28 = 64 players; then 32,16,8,4,2,132, 16, 8, 4, 2, 1 matches. Total 36+32+16+8+4+2+1=9936 + 32 + 16 + 8 + 4 + 2 + 1 = 99.

The trap

Getting tangled in the round-by-round structure (or halving 100 to get 50 first-round matches) instead of counting eliminations.

Common mistakes

  • Getting tangled in the round-by-round structure (or halving 100100 to get 5050 first-round matches) instead of counting eliminations.
  • Counting 100100 matches by forgetting that the champion is never eliminated, which gives "divisible by 2" and "by 5."

Techniques

Map the objects to something easier to count