HMMT 二月 2015 · 冲刺赛 · 第 11 题

英文解析

11. [ 6 ] Find
    √
     
     
    2000000
    ∞
    1 +
    ∑
     k 
     4
    ,
    k =02
    where ⌊ x ⌋ denotes the largest integer less than or equal to x .
    Answer: 1414 The k th floor (for k ≥ 0) counts the number of positive integer solutions tok 2 6 k 2
    4 (2 x − 1) ≤ 2 · 10 . So summing over all k , we want the number of integer solutions to 4 (2 x − 1) ≤
    2 · 10 with k ≥ 0 and x ≥ 1. But each positive integer can be uniquely represented as a power of 26
    √
    times an odd (positive) integer, so there are simply ⌊ 10 2 ⌋ = 1414 solutions.3