AMC 8 · 2016 · #18

Grade 4 countingarithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Multi-Segment Path or Count Sum

systematic-enumerationpattern-recognitionmulti-digit-arithmetic easier-related-problemidentify-subproblems ↑ Prerequisites: multi-digit-arithmeticpattern-recognition

📏 Short solution 💡 2 insights

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Problem

A track meet has 216 sprinters and a 6-lane track. In each race 6 runners compete; the 5 non-winners are eliminated and the winner runs again later. How many races are needed before exactly one champion is left?

Pick an answer.

(A)

$mbox{ }36$

(B)

$mbox{ }42$

(C)

$mbox{ }43$

(D)

$mbox{ }60$

(E)

$mbox{ }72$

AMC 8 2016 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Solve an Easier Related Problem

216 sprinters is a lot to picture, so Tool #9 (Easier Related Problem) starts with a tiny field — say 6 or 36 sprinters — to spot the underlying rule. The rule that pops out is simple: every race kills exactly 5 sprinters, and we need to kill everyone except the champion. From there, Tool #7 (Identify Subproblems) gives a clean cross-check: split the tournament into rounds and count the races in each round separately, then add.

1STEP 1

Tool #9: try the smallest full-track cases — 6 sprinters need 1 race, and 36 sprinters give 6 + 1 = 7 races.

36 sprinters → 6 + 1 = 7 races

2STEP 2

Count eliminations: for 36 sprinters, 35 lose over 7 races, and 7 × 5 = 35 — so every race removes exactly 5 sprinters.

races × 5 = eliminations

3STEP 3

Apply the rule to 216: everyone but the champion is out, so $\frac{216 - 1}{5}$ = $\frac{215}{5}$ = 43 races.

\frac{216 - 1}{5}

\frac{215}{5}

= 43 races

4STEP 4

Cross-check with Tool #7 by rounds: 216→36 races, 36→6 races, 6→1 race, so 36 + 6 + 1 = 43 → (C).

36 + 6 + 1 = 43 → (C)

Answer

mbox{ }43

Two independent methods give the same answer of 43, which sits between the trap answers (A) 36 (only Round 1) and (D) 60 (a careless 300 ÷ 5). Sanity check the eliminations: 43 races × 5 losers each = 215 eliminated, leaving exactly 1 champion out of 216. The arithmetic closes perfectly.

💡Key takeaway

Big tournament numbers shrink fast once you spot that every race eliminates exactly 5 runners — Grade 4 division finishes it.

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