HMMT 二月 2013 · 团队赛 · 第 4 题

英文解析

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
   ∑
   ⌊ 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.