Problem 18 · 2016 AMC 8
Medium
Logic & Word Problems
careful-counting
In an All-Area track meet, 216 sprinters enter a 100-meter dash competition. The track has 6 lanes, so only 6 sprinters can compete at a time. At the end of each race, the five non-winners are eliminated, and the winner will compete again in a later race. How many races are needed to determine the champion sprinter?
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Answer: C — 43 races.
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Hint 1 of 2
Flip the question. Instead of asking "how many races happen?" (which means tracking winners through messy rounds), ask "how many sprinters must DISAPPEAR?" That number is fixed, no matter how the bracket is arranged.
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Hint 2 of 2
This is the single-elimination principle: to crown ONE champion out of N, exactly N − 1 competitors must be eliminated — and here each race removes a fixed 5 of them. So races = (eliminations needed) ÷ (eliminated per race).
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Approach: count what must be eliminated, not the races themselves
- There's exactly one champion, so 216 − 1 = 215 sprinters must be eliminated — this total never changes, however the heats are organized.
- Each race eliminates exactly 5 (the non-winners), so the number of races = 215 ÷ 5 = 43.
- Worth keeping: any "games needed to find one winner" equals (players − 1) ÷ (losers per game). A 64-team single-elimination bracket needs 64 − 1 = 63 games, no bracket-drawing required.
- Sanity check: 215 isn't a multiple of 5? It is (215 = 5 × 43), and the final race must end with 1 winner from however many remain — the arithmetic working out cleanly confirms the setup.
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