HMMT 十一月 2011 · 冲刺赛 · 第 8 题
HMMT November 2011 — Guts Round — Problem 8
题目详情
- [ 7 ] Let a, b, c be not necessarily distinct integers between 1 and 2011, inclusive. Find the smallest ab + c possible value of . a + b + c
解析
- [ 7 ] Let a, b, c be not necessarily distinct integers between 1 and 2011, inclusive. Find the smallest ab + c possible value of . a + b + c 2 Answer: We have 3 ab + c ab − a − b = + 1 . a + b + c a + b + c ab − a − b We note that < 0 ⇔ ( a − 1)( b − 1) < 1, which only occurs when either a = 1 or b = 1. Without a + b + c loss of generality, let a = 1. Then, we have a value of − 1
- 1 . b + c + a 2 We see that this is minimized when b and c are also minimized (so b = c = 1), for a value of . 3