2003 AMC 12A Problem 7

How many non-congruent triangles with perimeter

7

have integer side lengths?

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Solution

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.

Concepts: triangle inequality · partitions and compositions

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