HMMT 十一月 2009 · GEN1 赛 · 第 2 题

英文解析

2. [ 3 ] Given that a + b + c = 5 and that 1 ≤ a, b, c ≤ 2, what is the minimum possible value of + ?
   a + b b + c
   Answer: If a > 1 and b < 2, we can decrease the sum by decreasing a and increasing b . You can 4
   follow a similar procedure if c > 1 and b < 2. Therefore, the sum is minimized when b = 2. We can 7
   a + c +4 7
   then cross-multiply the two fractions and see that we are trying to minimize = .
   ( a +2)( c +2) ( a +2)( c +2)
   The product of two numbers with a fixed sum is maximized when those two numbers are equal, so
   7 3 4
   is minimized for a = c = , which gives us an answer of .
   ( a +2)( c +2) 2 7