🇺🇸 AMC 8 ⇄ switch contest
2023 AMC 8

Problem 14

Problem 14 · 2023 AMC 8 Hard
Number Theory complementary-countingcareful-counting

Nicolas is planning to send a package to his friend Anton, who is a stamp collector. To pay for the postage, Nicolas would like to cover the package with a large number of stamps. Suppose he has a collection of 5-cent, 10-cent, and 25-cent stamps, with exactly 20 of each type. What is the greatest number of stamps Nicolas can use to make exactly $7.10 in postage?

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Answer: E — 55 stamps.
Show hints
Hint 1 of 2
‘Use the most stamps’ is hard to chase directly. Flip it: he owns a fixed pile worth a fixed total, so using the most stamps means setting aside the fewest. What's the whole pile worth?
Still stuck? Show hint 2 →
Hint 2 of 2
All 60 stamps total $8.00, and he only needs $7.10 — so he must hold back exactly $0.90. Now the easy question: what's the fewest stamps that make $0.90?
Show solution
Approach: minimize stamps removed, not maximize stamps used
  1. Maximizing what you use is the same as minimizing what you leave out — and since the whole pile is fixed, the leftover is fixed in value too. That flip turns a messy ‘maximize’ into a tidy ‘minimize.’
  2. All 60 stamps total 20 × ($0.05 + $0.10 + $0.25) = 20 × $0.40 = $8.00. To pay $7.10 he must hold back $8.00 − $7.10 = $0.90.
  3. Fewest stamps making $0.90: grab the biggest coins first — three 25¢ (75¢), one 10¢, one 5¢ = $0.90 with just 5 stamps.
  4. Stamps used = 60 − 5 = 55. This transfers: ‘use the most/fewest of a fixed supply’ problems are almost always easier solved by counting the complement — what you don't use.
Another way — build up directly (MAA):
  1. To pile on stamps, lean on the small ones and use as few 25¢ as possible. The 5¢ and 10¢ together max out at $1.00 + $2.00 = $3.00, so you need at least $4.10 more from quarters — at least seventeen 25¢ = $4.25.
  2. That leaves $7.10 − $4.25 = $2.85. Since $2.85 isn't a whole number of dimes, use an odd count of nickels: nineteen 5¢ = $0.95 leaves $1.90 = nineteen 10¢.
  3. Total: 17 + 19 + 19 = 55 stamps.
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