HMMT 十一月 2015 · 冲刺赛 · 第 2 题

英文解析

2. [ 5 ] The three sides of a right triangle form a geometric sequence. Determine the ratio of the length ofthe hypotenuse to the length of the shorter leg.
   Proposed by: Alexander Katz
   √
   1+ 5
   Answer:
   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 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
   r 222 problem as the ratio of the hypotenuse to the shorter side is = r .2