# 2014 AMC 12B Problems/Problem 12

## Problem

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? $\textbf{(A)}\ 8\qquad\textbf{(B)}\ 9\qquad\textbf{(C)}\ 10\qquad\textbf{(D)}\ 11\qquad\textbf{(E)}\ 12$

## Solution

Define $T$ to be the set of all integral triples $(a, b, c)$ such that $a \ge b \ge c$, $b+c > a$, and $a, b, c < 5$. Now we enumerate the elements of $T$: $(4, 4, 4)$ $(4, 4, 3)$ $(4, 4, 2)$ $(4, 4, 1)$ $(4, 3, 3)$ $(4, 3, 2)$ $(3, 3, 3)$ $(3, 3, 2)$ $(3, 3, 1)$ $(3, 2, 2)$ $(2, 2, 2)$ $(2, 2, 1)$ $(1, 1, 1)$

It should be clear that $|S|$ is simply $|T|$ minus the larger "duplicates" (e.g. $(2, 2, 2)$ is a larger duplicate of $(1, 1, 1)$). Since $|T|$ is $13$ and the number of higher duplicates is $4$, the answer is $13 - 4$ or $\boxed{\textbf{(B)}\ 9}$.

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 