HMMT 二月 2012 · 代数 · 第 7 题

HMMT February 2012 — Algebra — Problem 7

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 for x > 1.

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 for x > 1. √ k k Answer: y = 2 We note that ( a ⊗ b ) = ( a ⊗ b ) for all positive integers k . Then for all rational p p 1 p p q q q numbers we have a ⊗ b = ( a ⊗ b ) = ( a ⊗ b ) . So by continuity, for all real numbers a, b , it follows q a b ab ab log ( x ) log ( y ) 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 .