HMMT 二月 2003 · 代数 · 第 6 题

HMMT February 2003 — Algebra — Problem 6

Let a = 1, and let a = b n /a c for n > 1. Determine the value of a . 1 n n − 1 999 3 2 3 3 3

解析

Let a = 1, and let a = b n /a c for n > 1. Determine the value of a . 1 n n − 1 999 Solution: 999 We claim that for any odd n , a = n . The proof is by induction. To get the base n 3 3 cases n = 1 , 3, we compute a = 1, a = b 2 / 1 c = 8, a = b 3 / 8 c = 3. And if 1 2 3 3 2 the claim holds for odd n ≥ 3, then a = b ( n + 1) /n c = n + 3 n + 3, so a = n +1 n +2 2 n +3 n +2 3 2 3 2 2 b ( n +2) / ( n +3 n +3) c = b ( n +6 n +12 n +8) / ( n +3 n +2) c = b n +2+ c = n +2. 2 n +3 n +3 So the claim holds, and in particular, a = 999. 999 3 2 3 3 3