2014 AMC 12B Problem 12

A set

S

consists of triangles whose sides have integer lengths less than

5,

and no two elements of

S

are congruent or similar. What is the largest number of elements that

S

can have?

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Solution

Write each triangle by its side lengths in nonincreasing order. Only one equilateral triangle is allowed (all are similar), and of the similar pair

2,2,1

and

4,4,2

only one may appear. The remaining valid, pairwise non-similar triangles are

\begin{gathered} 4\,4\,3,\ 4\,4\,1,\ 4\,3\,3,\ 4\,3\,2,\ \\ 3\,3\,2,\ 3\,3\,1,\ 3\,2\,2, \end{gathered}

seven in all. Together with one equilateral and one of the similar pair,

S

has at most

9

elements. Thus, the correct answer is B.