HMMT 十一月 2017 · 冲刺赛 · 第 29 题

HMMT November 2017 — Guts Round — Problem 29

[ 19 ] Consider a sequence x such that x = x = 1, x = . Suppose that x = n 1 2 3 n 2 3 2 x − x x n − 1 n − 3 n − 2 1 for all n ≥ 4. Find the least n such that x ≤ . n 6 10 2

解析

[ 19 ] Consider a sequence x such that x = x = 1, x = . Suppose that x = n 1 2 3 n 2 3 2 x − x x n − 1 n − 3 n − 2 1 for all n ≥ 4. Find the least n such that x ≤ . n 6 10 Proposed by: Mehtaab Sawhney Answer: 13 x x x x n − 1 n − 3 n − 2 n − 1 The recursion simplifies to + = 2 . So if we set y = for n ≥ 2 then we have n x x x x n n − 2 n − 1 n y − y = y − y for n ≥ 3, which means that { y } is an arithmetic sequence. From the n n − 1 n − 1 n − 2 n n − 1 3 n 2 starting values we have y = 1 , y = , so y = for all n . (This means that x = .) Since 2 3 n n 2 2 n ! x 6 1 = y y · · · y , it suffices to find the minimal n such that the RHS is at least 10 . Note that 2 3 n x n 5 5 5 2 6 y y · · · y = 1 · (1 . 5 · 2 · 2 . 5 · 3 · 3 . 5) · (4 · 4 . 5 · 5 · 5 . 5 · 6) < 2 . 5 · 5 = 12 . 5 < 200 · 12 . 5 = 500000 < 10 , 2 3 12 while 6 y y · · · y = 1 · (1 . 5 · 2 · 2 . 5 · 3) · (3 . 5 · 4 · 4 . 5) · (5 · 5 . 5 · 6 · 6 . 5) > 20 · 60 · 900 = 1080000 > 10 , 2 3 13 so the answer is 13. 2