HMMT 十一月 2018 · 冲刺赛 · 第 24 题

HMMT November 2018 — Guts Round — Problem 24

[ 12 ] Let ABCD be a convex quadrilateral so that all of its sides and diagonals have integer lengths. ◦ Given that ∠ ABC = ∠ ADC = 90 , AB = BD , and CD = 41, find the length of BC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT November 2018, November 10, 2018 — GUTS ROUND Organization Team Team ID#

解析

[ 12 ] Let ABCD be a convex quadrilateral so that all of its sides and diagonals have integer lengths. Given that \ ABC = \ ADC = 90 , AB = BD , and CD = 41, find the length of BC . Proposed by: Anders Olsen Answer: 580 Let the midpoint of AC be O which is the center of the circumcircle of ABCD. ADC is a right triangle 2 2 2 with a leg of length 41, and 41 = AC AD = ( AC AD )( AC + AD ). As AC, AD are integers and 41 is prime, we must have AC = 840 , AD = 841. Let M be the midpoint of AD. 4 AOM ⇠ 4 ACD, so p 2 2 BM = BO + OM = 841 / 2 + 41 / 2 = 441 . Then AB = 420 + 441 = 609 (this is a 20-21-29 triangle p 2 2 2 2 2 scaled up by a factor of 21). Finally, BC = AC AB so BC = 841 609 = 580 .