HMMT 二月 2007 · 代数 · 第 8 题

英文解析

8. [ 6 ] Let A := Q \ { 0 , 1 } denote the set of all rationals other than 0 and 1. A function f : A → R hasthe property that for all x ∈ A ,
   ( )
   f ( x ) + f 1 − = log | x | .1
   Compute the value of f (2007).x
   Answer: log ( 2007 / 2006 ) . Let g : A → A be defined by g ( x ) := 1 − 1 /x ; the key property is thatg ( g ( g ( x ))) = 1 − = x.1
   1 −1
   1 −1
   The given equation rewrites as f ( x ) + f ( g ( x )) = log | x | . Substituting x = g ( y ) and x = g ( g ( z )) givesxthe further equations f ( g ( y )) + f ( g ( g ( y ))) = log | g ( x ) | and f ( g ( g ( z ))) + f ( z ) = log | g ( g ( x )) | . Settingy and z to x and solving the system of three equations for f ( x ) givesf ( x ) = · (log | x | − log | g ( x ) | + log | g ( g ( x )) | ) .1
   2006 − 12
   For x = 2007, we have g ( x ) = and g ( g ( x )) = , so that
   2007 2006
   ∣ ∣ ∣ ∣
   2006 − 1
   ∣ ∣ ∣ ∣
   log | 2007 | − log + log
   2007 2006
   f (2007) = = log (2007 / 2006) .
   4 32