HMMT 二月 2014 · 团队赛 · 第 4 题

英文解析

4. [ 20 ] Compute
   ⌊ ⌋
   100100
   ∑
   .2
   50 k
   2 + 2
   k =0
   (Here, if x is a real number, then ⌊ x ⌋ denotes the largest integer less than or equal to x .)
   2100
   Answer: 101 · 2 − 50 Let a = . Notice that, for k = 0 , 1 , . . . , 49,49
   k 50 k
   2 +2
   100 100 100 50+ k
   2 2 2 2
   a + a = + = + = 250
   k 100 − k
   50 k 50 100 − k 50 k k 50
   2 + 2 2 + 2 2 + 2 2 + 2
   It is clear that for k = 0 , 1 , . . . , 49, a , a ∈ / Z , so ⌊ a ⌋ + ⌊ a ⌋ = 2 − 1 (since the sum of floors 50
   k 100 − k k 100 − kis an integer less than a + a but greater than a − 1 + a − 1). Thus,
   k 100 − k k 100 − k
   ∑100
   50 49 49
   ⌊ a ⌋ = 50 · (2 − 1) + 2 = 101 · 2 − 50
   k =0 k