Difference between revisions of "2014 AMC 12B Problems/Problem 12"
(→Problem) |
(→Solution) |
||
Line 35: | Line 35: | ||
<math>(1, 1, 1)</math> | <math>(1, 1, 1)</math> | ||
− | It should be clear that <math>|S|</math> is simply <math>|T|</math> minus the larger "duplicates" (e.g. <math>(2, 2, 2)</math> is a larger duplicate of <math>(1, 1, 1)</math>). Since <math>|T|</math> is 13 and the number of higher duplicates is 4, the answer is <math>13 - 4</math> or <math>9 (B)</math>. | + | It should be clear that <math>|S|</math> is simply <math>|T|</math> minus the larger "duplicates" (e.g. <math>(2, 2, 2)</math> is a larger duplicate of <math>(1, 1, 1)</math>). Since <math>|T|</math> is <math>13</math> and the number of higher duplicates is <math>4</math>, the answer is <math>13 - 4</math> or <math>9 (B)</math>. |
Revision as of 21:31, 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 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 .