HMMT 十一月 2011 · 冲刺赛 · 第 10 题

HMMT November 2011 — Guts Round — Problem 10

[ 8 ] Determine the number of integers D such that whenever a and b are both real numbers with 2 2 − 1 / 4 < a, b < 1 / 4, then | a − Db | < 1.

解析

[ 8 ] Determine the number of integers D such that whenever a and b are both real numbers with 2 2 − 1 / 4 < a, b < 1 / 4, then | a − Db | < 1. Answer: 32 We have 2 2 a − 1 a + 1 2 2 − 1 < a − Db < 1 ⇒ < D < . 2 2 b b Guts Round 2 2 2 2 a − 1 . 25 − 1 a + 1 0 + 1 We have is maximal at − 15 = and is minimal at = 16. However, since 2 2 2 2 b . 25 b . 25 we cannot have a, b = ± . 25, checking border cases of -15 and 16 shows that both of these values are possible for D . Hence, − 15 ≤ D ≤ 16, so there are 32 possible values of D .