HMMT 十一月 2008 · 冲刺赛 · 第 9 题

HMMT November 2008 — Guts Round — Problem 9

[ 7 ] Let a = , and 0 7 { 1 2 a if a < n n 2 a = n +1 1 2 a − 1 if a ≥ . n n 2 Find a . 2008 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . st 1 HARVARD-MIT NOVEMBER TOURNAMENT, 8 NOVEMBER 2008 — GUTS ROUND 2

解析

[ 7 ] Let a = , and 0 7 { 1 2 a if a < n n 2 a = n +1 1 2 a − 1 if a ≥ . n n 2 Find a . 2008 5 Answer: We calculate the first few a : i 7 5 3 6 a = , a = , a = = a 1 2 3 0 7 7 7 6 5 So this sequence repeats every three terms, so a = a = . Then a = . 2007 0 2008 7 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . st 1 HARVARD-MIT NOVEMBER TOURNAMENT, 8 SATURDAY 2008 — GUTS ROUND 2