2003 AMC 10A Problem 7

How many non-congruent triangles with perimeter

7

have integer side lengths?

Show solution

Solution

The longest side cannot exceed

3,

since otherwise the other two sides could not reach it. The only possibilities are side lengths

1,

3,

3

and

2,

2,

3,

giving

2

triangles. Thus, the correct answer is B.

Concepts: triangle inequality · systematic listing

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.