AMC 10 · 2008 · #4

Grade 5 arithmetic

Archetype Pair-Sum Invariant └ Sub-archetype Universe Total Minus Known

optimization extremal-construction ↑ Prerequisites: multi-digit-arithmetic

📏 Short solution 💡 1 insight

Problem

A semipro baseball league has teams with 21 players each. Every player must be paid at least 15,000 dollars, and the total of all 21 salaries on one team cannot exceed 700,000 dollars. What is the maximum possible salary, in dollars, for a single player?

Pick an answer.

(A)

$\ 270,000$

(B)

$\ 385,000$

(C)

$\ 400,000$

(D)

$\ 430,000$

(E)

$\ 700,000$

AMC 10 2008 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Extreme Principle

The $700,000 cap is shared among all 21 players. To hand one player as much as possible, push everyone else to the smallest amount the rules allow. Once the other 20 sit at the floor, whatever is left of the cap belongs to the last player.

1STEP 1

Push the others to the minimum

The 21 salaries share one fixed cap, so drop the other 20 players to the league minimum of 15,000 dollars each.

one player = 700,000 - (salaries of the other 20)

2STEP 2

Cost of the other 20 players

Twenty players at the 15,000 minimum cost 20 times 15,000, a floor of 300,000 dollars.

20 × 15,000 = 300,000

3STEP 3

Whatever is left goes to one player

Take that 300,000 out of the 700,000 cap and 400,000 dollars is left for the one player — choice (C).

700,000 - 300,000 = 400,000

Answer

400,000

Check the total: one player at 400,000 plus twenty players at 15,000 gives 400,000 + 300,000 = 700,000, exactly the cap and not over it. Every salary is at least 15,000, so all rules hold. Choice (E) 700,000 fails because it leaves nothing for the other 20 players, who still each need 15,000 dollars.

💡Key takeaway

When a total is capped, make one share the biggest by shrinking every other share to its minimum.

Push the others to the minimum
Cost of the other 20 players
Whatever is left goes to one player

Push the others to the minimum

Cost of the other 20 players

Whatever is left goes to one player

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