HMMT 十一月 2012 · 冲刺赛 · 第 34 题

HMMT November 2012 — Guts Round — Problem 34

[ 20 ] For a positive integer n , let τ ( n ) be the number of divisors of n . Determine τ ( n ). Your n =1 { ⌊ ⌋} ( ) 3 | S − k | score will be max 0 , 20 1 − , where k is your answer and S is the actual answer. S

解析

[ 20 ] Answer: 15612 Note that the number of integers between 1 and 2012 that have n as a divi- 2012 sor is ⌊ ⌋ . Therefore, if we sum over the possible divisors, we see that the sum is equivalent to n ∑ ∑ ∑ 2012 2012 2012 2012 2012 1 ⌊ ⌋ . This can be approximated by = 2012 ≈ 2012 ln(2012). As it turns d =1 d =1 d =1 d d d out, 2012 ln(2012) ≈ 15300, which is worth 18 points. Using the very rough approximation ln(2012) ≈ 7 still gives 14 points.