2016 AIME II Problems/Problem 5

Triangle $ABC_0$ has a right angle at $C_0$ . Its side lengths are pariwise relatively prime positive integers, and its perimeter is $p$ . Let $C_1$ be the foot of the altitude to $\overline{AB}$ , and for $n \geq 2$ , let $C_n$ be the foot of the altitude to $\overline{C_{n-2}B}$ in $\triangle C_{n-2}C_{n-1}B$ . The sum $\sum_{i=1}^\infty C_{n-2}C_{n-1} = 6p$ . Find $p$ .

Solution

Note that by counting the area in 2 ways, the first altitude is $\dfrac{ac}{b}$ . By similar triangles, the common ratio is $\dfrac{a}{c}$ for reach height, so by the geometric series formula, we have $6p=\dfrac{\dfrac{ac}{b}}{1-\dfrac{a}{c}}$ . Testing triangles of the form $2p+1, 2p^{2}+2p, 2p^{2}+2p+1$ , we have $13, 84, 85$ satisfy this equation, so $p=13+84+85=\boxed{182}$ .

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2016 AIME II Problems/Problem 5

Solution