2003 AMC 12A Problem 7

Attempt Problem 7 of the 2003 AMC 12A below, then check your answer against the professionally curated solution 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 problems are used with official legal permission of the Mathematical Association of America (MAA).

How many non-congruent triangles with perimeter

7

have integer side lengths?

1

2

3

4

5

The longest side must be less than half the perimeter

The longest side is at most

3

Let the sides be

a\le b\le c

with

a+b+c=7.

The triangle inequality requires

c\lt a+b,

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.

Problem 7 in other years