HMMT 二月 2007 · 代数 · 第 8 题

HMMT February 2007 — Algebra — Problem 8

[ 6 ] Let A := Q \ { 0 , 1 } denote the set of all rationals other than 0 and 1. A function f : A → R has the property that for all x ∈ A , ( ) 1 f ( x ) + f 1 − = log | x | . x Compute the value of f (2007). 4 3

解析

[ 6 ] Let A := Q \ { 0 , 1 } denote the set of all rationals other than 0 and 1. A function f : A → R has the property that for all x ∈ A , ( ) 1 f ( x ) + f 1 − = log | x | . x Compute the value of f (2007). Answer: log ( 2007 / 2006 ) . Let g : A → A be defined by g ( x ) := 1 − 1 /x ; the key property is that 1 g ( g ( g ( x ))) = 1 − = x. 1 1 − 1 1 − x The given equation rewrites as f ( x ) + f ( g ( x )) = log | x | . Substituting x = g ( y ) and x = g ( g ( z )) gives the further equations f ( g ( y )) + f ( g ( g ( y ))) = log | g ( x ) | and f ( g ( g ( z ))) + f ( z ) = log | g ( g ( x )) | . Setting y and z to x and solving the system of three equations for f ( x ) gives 1 f ( x ) = · (log | x | − log | g ( x ) | + log | g ( g ( x )) | ) . 2 2006 − 1 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) . 2 4 3