返回题库

HMMT 二月 2014 · 代数 · 第 3 题

HMMT February 2014 — Algebra — Problem 3

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

Let
A=1
6
(log2(3))3(log2(6))3(log2(12))3+ (log2(24))3
.
Compute 2A.

解析

英文解析

  1. Let
    ( )
    3 3 3 31
    A = (log (3)) − (log (6)) − (log (12)) + (log (24)) .
    2 2 2 2
    A6
    Compute 2 .
    a 3 3 3 31
    Answer: 72 Let a = log (3), so 2 = 3 and A = [ a − ( a + 1) − ( a + 2) + ( a + 3) ]. But
    1 162
    3 3 2 2 2
    ( x + 1) − x = 3 x + 3 x + 1, so A = [3( a + 2) + 3( a + 2) − 3 a − 3 a ] = [4 a + 4 + 2] = 2 a + 3. Thus
    6 2
    A a 2 3
    2 = (2 ) (2 ) = 9 · 8 = 72. 2