HMMT 二月 2006 · 冲刺赛 · 第 16 题

HMMT February 2006 — Guts Round — Problem 16

[8] A sequence a , a , a , . . . of positive reals satisfies a = . Determine all a such 1 2 3 n +1 1 2 √ √ 6+ 2 that a = for some positive integer i . i 4

解析

A sequence a , a , a , . . . of positive reals satisfies a = . Determine all a such 1 2 3 n +1 1 2 √ √ 6+ 2 that a = for some positive integer i . i 4 √ √ √ 2 + 6 3 1 Answer: , , 2 2 2 Solution: Clearly a < 1, or else 1 ≤ a ≤ a ≤ a ≤ . . . . We can therefore 1 1 2 3 √ θ 1 + cos θ ◦ ◦ write a = cos θ for some 0 < θ < 90 . Note that cos = , and cos 15 = 1 2 2 4 √ √ 6 + 2 ◦ ◦ ◦ . Hence, the possibilities for a are cos 15 , cos 30 , and cos 60 , which are 1 4 √ √ √ 2 + 6 3 1 , , and . 2 2 2