HMMT 十一月 2012 · 冲刺赛 · 第 34 题
HMMT November 2012 — Guts Round — Problem 34
题目详情
英文原题
- [ 20 ] For a positive integer n , let τ ( n ) be the number of divisors of n . Determine τ ( n ). Yourn =1
{ ⌊ ⌋}
( )
| S − k |3
score will be max 0 , 20 1 − , where k is your answer and S is the actual answer. S
解析
英文解析
- [ 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 ton
∑ ∑ ∑
2012 2012 2012
2012 2012 1
⌊ ⌋ . This can be approximated by = 2012 ≈ 2012 ln(2012). As it turnsd =1 d =1 d =1
d d dout, 2012 ln(2012) ≈ 15300, which is worth 18 points. Using the very rough approximation ln(2012) ≈ 7
still gives 14 points.