2023 AMC 8 Problem 14
Attempt Problem 14 of the 2023 AMC 8 below, then check your answer against the video solution and professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2023 AMC 8 solutions, or check the answer key.
All problems are used with official legal permission of the Mathematical Association of America (MAA).
14.
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 -cent, -cent, and -cent stamps, with exactly of each type. What is the greatest number of stamps Nicolas can use to make exactly in postage?
(Note: The amount corresponds to dollars and cents. One dollar is worth cents.)
Answer: E
Small hint:
Use as many low-value stamps as possible
Big hint:
All - and -cent stamps total cents, leaving a nonmultiple of
Video solution:
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Written solution:
Let be the numbers of -, -, and -cent stamps, and let Dividing the value equation by gives so
Suppose Then Also so Combining these inequalities gives but then a contradiction. Thus at most stamps can be used.
The bound is attainable with Hence the greatest possible number of stamps is
Thus, E is the correct answer.
Problem 14 in other years
- 1985 AMC 8
- 1986 AMC 8
- 1987 AMC 8
- 1988 AMC 8
- 1989 AMC 8
- 1990 AMC 8
- 1991 AMC 8
- 1992 AMC 8
- 1993 AMC 8
- 1994 AMC 8
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- 1998 AMC 8
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