HMMT 二月 2013 · 团队赛 · 第 4 题

HMMT February 2013 — Team Round — Problem 4

[ 20 ] Let a , a , a , a , a be real numbers whose sum is 20. Determine with proof the smallest possible 1 2 3 4 5 value of X b a + a c . i j 1  i<j  5

解析

[ 20 ] Let a , a , a , a , a be real numbers whose sum is 20. Determine with proof the smallest possible 1 2 3 4 5 value of ∑ ⌊ a + a ⌋ . i j 1 ≤ i<j ≤ 5 Answer: 72 We claim that the minimum is 72. This can be achieved by taking a = a = a = 1 2 3 a = 0 . 4 and a = 18 . 4. To prove that this is optimal, note that 4 5 ∑ ∑ ∑ ⌊ a + a ⌋ = ( a + a ) − { a + a } = 80 − { a + a } , i j i j i j i j 1 ≤ i<j ≤ 5 1 ≤ i<j ≤ 5 1 ≤ i<j ≤ 5 so it suffices to maximize 5 5 ∑ ∑ ∑ { a + a } = { a + a } + { a + a } , i j i i +2 i i +1 1 ≤ i<j ≤ 5 i =1 i =1 where a = a and a = a , Taking each sum modulo 1, it is clear that both are integers. Thus, the 6 1 7 2 above sum is at most 2 · 4 = 8, and our original expression is at least 80 − 8 = 72, completing the proof.