HMMT 二月 2017 · 冲刺赛 · 第 21 题

HMMT February 2017 — Guts Round — Problem 21

[ 12 ] Let P and A denote the perimeter and area respectively of a right triangle with relatively prime 2 P integer side-lengths. Find the largest possible integral value of A

解析

[ 12 ] Let P and A denote the perimeter and area respectively of a right triangle with relatively prime 2 P integer side-lengths. Find the largest possible integral value of A Proposed by: Sam Korsky Assume WLOG that the side lengths of the triangle are pairwise coprime. Then they can be written 2 2 2 2 as m − n , 2 mn, m + n for some coprime integers m and n where m > n and mn is even. Then we obtain 2 P 4 m ( m + n ) = A n ( m − n ) But n, m − n, m, m + n are all pairwise coprime so for this to be an integer we need n ( m − n ) | 4 and by checking each case we find that ( m, n ) = (5 , 4) yields the maximum ratio of 45 .