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HMMT 二月 2004 · GEN1 赛 · 第 3 题

HMMT February 2004 — GEN1 Round — Problem 3

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. Suppose f is a function that assigns to each real number x a value f ( x ), and supposethe equationf ( x + x + x + x + x ) = f ( x ) + f ( x ) + f ( x ) + f ( x ) + f ( x ) − 8
    1 2 3 4 5 1 2 3 4 5
    holds for all real numbers x , x , x , x , x . What is f (0)?
    1 2 3 4 5
解析

英文解析

  1. Suppose f is a function that assigns to each real number x a value f ( x ), and supposethe equationf ( x + x + x + x + x ) = f ( x ) + f ( x ) + f ( x ) + f ( x ) + f ( x ) − 8
    1 2 3 4 5 1 2 3 4 5
    holds for all real numbers x , x , x , x , x . What is f (0)?
    1 2 3 4 5
    Solution: 2
    Plug in x = x = x = x = x = 0. Then the equation reads f (0) = 5 f (0) − 8, so
    1 2 3 4 5
    4 f (0) = 8, so f (0) = 2.