Difference between revisions of "2014 AMC 12B Problems/Problem 12"
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| − | Define <math>T</math> to be the set of all integral triples <math>(a, b, c)</math> such that <math>a \ge b \ge c</math>, <math>b+c > a</math>, and <math>a, b, c | + | Define <math>T</math> to be the set of all integral triples <math>(a, b, c)</math> such that <math>a \ge b \ge c</math>, <math>b+c > a</math>, and <math>a, b, c < 5</math>. Now we enumerate the elements of <math>T</math>: |
<math>(4, 4, 4)</math> | <math>(4, 4, 4)</math> | ||
Revision as of 22:01, 20 February 2014
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 
.
- Based on the wording of Problem 13 to specifically exclude triangles with zero area: "... triangle with positive area", the definition of a triangle in this test includes degenerate ones. That is, the triangle inequality is not strict. The following are possible degenerate triangles (excluding duplicates):
As the specifics to the definition of the triangle were not provided, and the only evidence of such (Problem 13) includes degenerates, we must assume the most general case and include these. Our final answer is then 
or 
.
