<p>Triangle <img alt="$ABC_0$" height="15" src="https://latex.artofproblemsolving.com/5/5/2/5524c593ef0a23285864f939ee5c0c1dbe84967f.png" width="47" /> has a right angle at <img alt="$C_0$" height="15" src="https://latex.artofproblemsolving.com/7/c/e/7ce4664de7db214d4c56a42b1922cbd0fb7e7785.png" width="19" />. Its side lengths are pairwise relatively prime positive integers, and its perimeter is <img alt="$p$" height="11" src="https://latex.artofproblemsolving.com/3/6/f/36f73fc1312ee0349b3f3a0f3bd9eb5504339011.png" width="10" />. Let <img alt="$C_1$" height="15" src="https://latex.artofproblemsolving.com/7/0/b/70b97eb4ad6e4e7a5a97e2547d9ed6e965960d0c.png" width="18" /> be the foot of the altitude to <img alt="$\overline{AB}$" height="18" src="https://latex.artofproblemsolving.com/8/4/0/840e2b592390eb6ec918fa6f3292716ce170de66.png" width="30" />, and for <img alt="$n \geq 2$" height="14" src="https://latex.artofproblemsolving.com/7/2/2/72287b09043ed45b372a517d764e12d07aff8071.png" width="43" />, let <img alt="$C_n$" height="15" src="https://latex.artofproblemsolving.com/4/e/1/4e173907ca992482385a50c6f62ede920362b1bf.png" width="21" /> be the foot of the altitude to <img alt="$\overline{C_{n-2}B}$" height="18" src="https://latex.artofproblemsolving.com/5/8/5/585a0e7f1f26e077fd69305d7fc214869f1b79bf.png" width="54" /> in <img alt="$\triangle C_{n-2}C_{n-1}B$" height="16" src="https://latex.artofproblemsolving.com/c/7/8/c78bfcb2119dd071cca669bf8e9da7d0ab31f2e0.png" width="108" />. The sum <img alt="$\sum_{n=2}^\infty C_{n-2}C_{n-1} = 6p$" height="48" src="https://latex.artofproblemsolving.com/8/7/2/87244e500055f0e66ecc76d9fd4f4439c7ef1e86.png" width="149" />. Find <img alt="$p$" height="11" src="https://latex.artofproblemsolving.com/3/6/f/36f73fc1312ee0349b3f3a0f3bd9eb5504339011.png" width="10" />.</p>

Question No: 368

Triangle $ABC_0$ has a right angle at $C_0$ . Its side lengths are pairwise 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 $\sum_{n=2}^\infty C_{n-2}C_{n-1} = 6p$ . Find $p$ .

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Tags: Algebra Geometry

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