HMMT 十一月 2011 · 冲刺赛 · 第 10 题
HMMT November 2011 — Guts Round — Problem 10
题目详情
英文原题
[8] Determine the number of integers Dsuch that whenever aandbare both real numbers with −1/4< a, b < 1/4, then |a2−Db2|<1.
解析
英文解析
- [ 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 arepossible for D . Hence, − 15 ≤ D ≤ 16, so there are 32 possible values of D .