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2014 AMC 12B Problems/Problem 12

Revision as of 21:15, 20 February 2014 by Mathgreen (talk | contribs) (→‎Solution)
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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 $T$:

$(5, 5, 5)$

$(5, 5, 4)$

$(5, 5, 3)$

$(5, 5, 2)$

$(5, 5, 1)$

$(5, 4, 4)$

$(5, 4, 3)$

$(5, 4, 2)$

$(5, 3, 3)$

$(4, 4, 4)$

$(4, 4, 3)$

$(4, 4, 2)$

$(4, 4, 1)$

$(4, 3, 3)$

$(4, 3, 2)$

$(3, 3, 3)$

$(3, 3, 2)$

$(3, 3, 1)$

$(3, 2, 2)$

$(2, 2, 2)$

$(2, 2, 1)$

$(1, 1, 1)$

It should be clear that $|S|$ is simply $|T| - t$, where $x$ 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, $x$ is

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