1956 AMC 12 Problem 43

Question

The number of scalene triangles having all sides of integral lengths, and perimeter less than

13

is:

1

2

3

4

18

Order the distinct integer sides as

a\lt b\lt c

and use

a+b\gt c

List possibilities by the largest side; the perimeter bound leaves only small values of

c

Write the distinct integer sides in increasing order. Checking the small possibilities under the triangle inequality and perimeter bound gives

\begin{aligned} &(2,3,4),\ (2,4,5),\\ &(3,4,5). \end{aligned}

For

c\ge6,

the smallest new scalene candidate satisfying

a+b\gt c

is

(3,4,6),

whose perimeter is already

13,

and larger choices cannot qualify. Thus there are

3

triangles.

The correct answer is C.

Problem 43 in other years