HMMT 二月 2006 · 冲刺赛 · 第 16 题
HMMT February 2006 — Guts Round — Problem 16
题目详情
英文原题
- [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 .
4 i
解析
英文解析
- 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 .
4 i
√ √ √
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 =
2 21
√ √4
6 + 2
° ° °
. Hence, the possibilities for a are cos 15 , cos 30 , and cos 60 , which are
√ √ √41
2 + 6 3 1
, , and .
2 2 2