HMMT 二月 2012 · 冲刺赛 · 第 34 题

HMMT February 2012 — Guts Round — Problem 34

[ 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 ⌊ ⌋ 23 be min(23 , ). Otherwise, your score will be 0. 3( U − L ) ∏ 2012 k

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 ⌊ ⌋ 23 be min(23 , ). Otherwise, your score will be 0. 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 is k =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