HMMT 二月 2012 · 代数 · 第 7 题
HMMT February 2012 — Algebra — Problem 7
题目详情
英文原题
Let ⊗be a binary operation that takes two positive real numbers and returns a pos itive real number. Suppose further that ⊗is continuous, commutative ( a⊗b=b⊗a), distributive across multiplication (a⊗(bc) = ( a⊗b)(a⊗c)), and that 2 ⊗2 = 4. Solve the equation x⊗y=xforyin terms of xfor x >1.
解析
英文解析
- Let ⊗ be a binary operation that takes two positive real numbers and returns a positive real number.
Suppose further that ⊗ is continuous, commutative ( a ⊗ b = b ⊗ a ), distributive across multiplication
( a ⊗ ( bc ) = ( a ⊗ b )( a ⊗ c )), and that 2 ⊗ 2 = 4. Solve the equation x ⊗ y = x for y in terms of x forx > 1.
√
k k
Answer: y = 2 We note that ( a ⊗ b ) = ( a ⊗ b ) for all positive integers k . Then for all rationalp pp 1
q q qpnumbers we have a ⊗ b = ( a ⊗ b ) = ( a ⊗ b ) . So by continuity, for all real numbers a, b , it followsa b ab ab log ( x ) log ( y )q
2 2
that 2 ⊗ 2 = (2 ⊗ 2) = 4 . Therefore given positive reals x, y , we have x ⊗ y = 2 ⊗ 2 =
log ( x ) log ( y )
2 2
4 .
log ( x ) log ( y ) 2 log ( x ) log ( y ) 2
2 2 2 2
If x = 4 = 2 then log ( x ) = 2 log ( x ) log ( y ) and 1 = 2 log ( y ) = log ( y ).
2 2 2 2 2
√
Thus y = 2 regardless of x .