HMMT 二月 2012 · 冲刺赛 · 第 34 题
HMMT February 2012 — Guts Round — Problem 34
题目详情
英文原题
[23] Let Q be the product of the sizes of all the non-empty subsets of {1,2, . . . , 2012}, and let M= log2(log2(Q)). Give lower and upper bounds LandUforM. If 0 < L≤M≤U, then your score will be min(23 ,/floorleftBig 23 3(U−L)/floorrightBig ). Otherwise, your score will be 0.
解析
英文解析
- [ 23 ] Let Q be the product of the sizes of all the non-empty subsets of { 1 , 2 , . . . , 2012 } , and let M =
log (log ( Q )). Give lower and upper bounds L and U for M . If 0 < L ≤ M ≤ U , then your score will
2 2
⌊ ⌋
be min(23 , ). Otherwise, your score will be 0.23
3( U − L )
Answer: 2015.318180... In this solution, all logarithms will be taken in base 2. It is clear that
( ) ( )
∑ ∑
2012 2011
2012 2012
log( Q ) = log( k ). By paring k with 2012 − k , we get 0 . 5 ∗ log( k (2012 − k )) +
k =1 k k =1 k
( ) ( )
∑ ∑
2012 2012
2012 2012
log(2012), which is between 0 . 5 ∗ log(2012) and log(2012) ; i.e., the answer isk =0 k k =0 k
2011 2012
between log(2012)2 and log(2012)2 . Thus log(log( Q )) is between 2011 + log(log(2012)) and
2012 + log(log(2012)). Also 3 < log(log(2012)) < 4. So we get 2014 < M < 2016.
∏
2012 k