2015 AMC 12B Problem 10

Attempt Problem 10 of the 2015 AMC 12B 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 2015 AMC 12B solutions, or check the answer key.

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10.

How many noncongruent integer-sided triangles with positive area and perimeter less than

15

are neither equilateral, isosceles, nor right triangles?

3

4

5

6

7

With sides

a \lt b \lt c,

the triangle inequality gives perimeter

\gt 2c

This forces

c \le 6;

list scalene triples and drop the right triangle

3, 4, 5

Let the distinct sides be

a \lt b \lt c.

Since

a + b \gt c,

the perimeter exceeds

2c,

2c \lt 15

and

c \le 6.

The scalene triples with perimeter less than

15

are

(6,5,3),

(6,5,2),

(6,4,3),

(5,4,3),

(5,4,2),

and

(4,3,2).

Of these, only

(5,4,3)

is a right triangle, leaving

5.

Thus, the correct answer is C.

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