HMMT 十一月 2015 · 冲刺赛 · 第 2 题

HMMT November 2015 — Guts Round — Problem 2

[ 5 ] The three sides of a right triangle form a geometric sequence. Determine the ratio of the length of the hypotenuse to the length of the shorter leg.

解析

[ 5 ] The three sides of a right triangle form a geometric sequence. Determine the ratio of the length of the hypotenuse to the length of the shorter leg. Proposed by: Alexander Katz √ 1+ 5 Answer: 2 Let the shorter leg have length , and the common ratio of the geometric sequence be r > 1. Then the 2 length of the other leg is `r , and the length of the hypotenuse is `r . Hence, 2 2 2 2 ` + ( `r ) = (r ) 2 2 2 4 = ⇒ ` ( r + 1) = ` r 2 4 = ⇒ r + 1 = r √ √ 1 + 5 4 2 2 1 ± 5 2 Hence, r − r − 1 = 0, and therefore r = . As r > 1, we have r = , completing the 2 2 2 r 2 problem as the ratio of the hypotenuse to the shorter side is = r .