HMMT 十一月 2009 · GEN1 赛 · 第 2 题

HMMT November 2009 — GEN1 Round — Problem 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

解析

[ 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 4 Answer: If a > 1 and b < 2, we can decrease the sum by decreasing a and increasing b . You can 7 follow a similar procedure if c > 1 and b < 2. Therefore, the sum is minimized when b = 2. We can 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