2014 AMC 12B Problems/Problem 12

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 ... ...

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2014 AMC 12B Problems/Problem 12

Solution