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HMMT 二月 2008 · 代数 · 第 4 题

HMMT February 2008 — Algebra — Problem 4

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [ 4 ] The function f satisfiesf ( x ) + f (2 x + y ) + 5 xy = f (3 x − y ) + 2 x + 12
    for all real numbers x, y . Determine the value of f (10). 3
解析

英文解析

  1. [ 4 ] The function f satisfiesf ( x ) + f (2 x + y ) + 5 xy = f (3 x − y ) + 2 x + 12
    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 whichwe get f (10) = − 49.
    x 1
    Remark: By setting y = , we see that the function is f ( x ) = − x + 1, and it can be checked that 2
    2 2
    this function indeed satisfies the given equation. 3