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?
$\textbf{(A)}\ 8\qquad\textbf{(B)}\ 9\qquad\textbf{(C)}\ 10\qquad\textbf{(D)}}\ 11\qquad\textbf{(E)}\ 12$ (Error compiling LaTeX. Unknown error_msg)
Solution
Define to be the set of all integral triples
such that
,
, and
. Now we enumerate the elements of
:
It should be clear that is simply
minus the larger "duplicates" (e.g.
is a larger duplicate of
). Since
is
and the number of higher duplicates is
, the answer is
or
.