2003 AMC 12A Problem 7 — Solution | LIVE by Po-Shen Loh

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2003 AMC 12A Problem 7

Below is the professionally curated solution for Problem 7 of the 2003 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2003 AMC 12A solutions, or check the answer key.

All of the real AMC 8, AMC 10, AMC 12, and AIME problems in our complete solution collection are used with official legal permission of the Mathematical Association of America (MAA).

Concepts:triangle inequality partitions and compositions

Difficulty rating: 1270

7.

How many non-congruent triangles with perimeter $7$ have integer side lengths?

$1$

$2$

$3$

$4$

$5$

Solution:

Let the sides be $a\le b\le c$ with $a+b+c=7.$ The triangle inequality requires $c\lt a+b,$ so $c\lt3.5,$ forcing $c=3.$

Then $a+b=4$ with $a\le b\le3,$ giving the triangles $1\text{-}3\text{-}3$ and $2\text{-}2\text{-}3.$

Thus, the correct answer is B.

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Problem 7 in Other Years

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