AMC 10 · 2003 · #15

Grade 4 number-theory
Archetype Pair-Sum Invariant └ Sub-archetype Universe Total Minus Known
complementary-countingdivisibility-rules complementary-countingeasier-related-problem ↑ Prerequisites: divisibility-rules
📏 Medium solution 💡 2 insights
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Problem
A single-elimination tournament starts with 100 players, and a player who loses a match is eliminated. In the first round 28 players get a bye and the remaining 72 players are paired off, and the rounds continue until exactly one player is left unbeaten. Find the total number of matches played, then decide whether that number is a prime number, or divisible by 2, 5, 7, or 11.

Pick an answer.

(A)
a prime number
(B)
divisible by 2
(C)
divisible by 5
(D)
divisible by 7
(E)
divisible by 11

AMC 10 2003 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Change Focus / Count the Complement

Counting matches round by round means tracking byes and how many players survive each round — fiddly and easy to slip on. Tool #16 (Change Focus) swaps the question: instead of counting matches, count the players who get eliminated, because each match removes exactly one player, so matches and eliminations are the same number. Tool #9 (Solve an Easier Related Problem) then makes the count trivial — to leave 1 champion out of 100 players, exactly 99 must be eliminated, and the bye numbers become irrelevant. Finally tool #3 (Eliminate Possibilities) checks 99 against the five descriptions to pick the one that fits.

1STEP 1

Match each match to one loser

Every match ends with exactly one loser who is out, so matches pair one-to-one with eliminations.

2STEP 2

Count how many get eliminated

Only the champion never loses, so 100 - 1 = 99 players go out — that means 99 matches, byes irrelevant.

100 - 1 = 99 matches
3STEP 3

Test 99 against the choices

Factor 99 = 3 × 3 × 11: composite, odd, not ending in 0 or 5, and 7 leaves a remainder — only 11 divides it, so (E).

99 = 3² × 11 → 11 ∣ 99 → (E)
Answer
divisible by 11
A quick sanity check by rounds confirms 99: round one plays 36 matches (the 72 paired players), leaving 28 byes + 36 winners = 64 players; then 32 + 16 + 8 + 4 + 2 + 1 = 63 more matches to whittle 64 down to 1. Total 36 + 63 = 99, exactly matching the elimination count. Since 99 = 9 × 11 is divisible by 11 and by no other option, (E) is the only description that fits.
💡Key takeaway

To find the number of matches, don't count games — count who gets knocked out: one champion means the other 99 players each lost once, so 99 matches.

  • Match each match to one loser
  • Count how many get eliminated
  • Test 99 against the choices

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