HMMT 十一月 2011 · 冲刺赛 · 第 8 题

英文解析

8. [ 7 ] Let a, b, c be not necessarily distinct integers between 1 and 2011, inclusive. Find the smallestab + cpossible value of .
   a + b + c
   Answer: We have 2
   ab + c ab − a − b 3 = + 1 .
   a + b + c a + b + cab − a − b
   We note that < 0 ⇔ ( a − 1)( b − 1) < 1, which only occurs when either a = 1 or b = 1. Withouta + b + closs of generality, let a = 1. Then, we have a value of − 1 + 1 .
   b + c + a
   We see that this is minimized when b and c are also minimized (so b = c = 1), for a value of .2 3