HMMT 二月 2004 · GEN1 赛 · 第 3 题
HMMT February 2004 — GEN1 Round — Problem 3
题目详情
- Suppose f is a function that assigns to each real number x a value f ( x ), and suppose the equation f ( 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
- Suppose f is a function that assigns to each real number x a value f ( x ), and suppose the equation f ( 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.