2023 AMC 10A Problem 4

Below is the professionally curated solution for Problem 4 of the 2023 AMC 10A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2023 AMC 10A solutions, or check the answer key.

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Concepts:triangle inequality bounding to limit cases

Difficulty rating: 1130

A quadrilateral has all integer side lengths, a perimeter of $26,$ and one side of length $4.$ What is the greatest possible length of one side of this quadrilateral?

$9$

$10$

$11$

$12$

$13$

Solution:

In any quadrilateral each side is shorter than the sum of the other three. Call the longest side $s.$ The rest sum to $26 - s,$ so $s \lt 26 - s,$ which gives $s \lt 13$ and hence $s \le 12.$ Can we hit $12?$ The sides $4, 12, 9, 1$ work, since $12 \lt 4 + 9 + 1.$ So the greatest length is $12.$ Therefore, the answer is D.

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