HMMT 二月 2008 · 代数 · 第 4 题
HMMT February 2008 — Algebra — Problem 4
题目详情
- [ 4 ] The function f satisfies 2 f ( x ) + f (2 x + y ) + 5 xy = f (3 x − y ) + 2 x + 1 for all real numbers x, y . Determine the value of f (10). 3
解析
- [ 4 ] The function f satisfies 2 f ( x ) + f (2 x + y ) + 5 xy = f (3 x − y ) + 2 x + 1 for all real numbers x, y . Determine the value of f (10). Answer: − 49 Setting x = 10 and y = 5 gives f (10) + f (25) + 250 = f (25) + 200 + 1, from which we get f (10) = − 49. x 1 2 Remark: By setting y = , we see that the function is f ( x ) = − x + 1, and it can be checked that 2 2 this function indeed satisfies the given equation. 3