Difference between revisions of "2014 AMC 12B Problems/Problem 12"

Revision as of 21:20, 20 February 2014

Solution

Define $T$ to be the set of all triples $(a, b, c)$ such that $a \ge b \ge c$ , $b+c > a$ , and $a, b, c \le 5$ . 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 $|T| - t$ , where $t$ is the number of triples $(d, e, f)$ such that there exists at least one triple $(kd, ke, kf)$ where $k \ge 1$ and $k \in \mathbb{N}$ . So, $t$ is... and the answer is ... ...

@@ Line 2: / Line 2: @@
 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>:
-<math>(5, 5, 5)</math>
-<math>(5, 5, 4)</math>
-<math>(5, 5, 3)</math>
-<math>(5, 5, 2)</math>
-<math>(5, 5, 1)</math>
-<math>(5, 4, 4)</math>
-<math>(5, 4, 3)</math>
-<math>(5, 4, 2)</math>
-<math>(5, 3, 3)</math>
 <math>(4, 4, 4)</math>

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Difference between revisions of "2014 AMC 12B Problems/Problem 12"

Revision as of 21:20, 20 February 2014

Solution