Difference between revisions of "2014 AMC 12B Problems/Problem 12"
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Define <math>T</math> to be the set of all 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 \le 5</math>. Now we enumerate the elements of <math>T</math>: | Define <math>T</math> to be the set of all 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 \le 5</math>. Now we enumerate the elements of <math>T</math>: | ||
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<math>(4, 4, 4)</math> | <math>(4, 4, 4)</math> |
Revision as of 22:20, 20 February 2014
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
, where
is the number of triples
such that there exists at least one triple
where
and
. So,
is... and the answer is ... ...