返回题库

HMMT 十一月 2016 · 冲刺赛 · 第 22 题

HMMT November 2016 — Guts Round — Problem 22

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [ 12 ] Let the function f : Z → Z take only integer inputs and have integer outputs. For any integers xand y , f satisfiesf ( x ) + f ( y ) = f ( x + 1) + f ( y − 1)
    If f (2016) = 6102 and f (6102) = 2016, what is f (1)?
解析

英文解析

  1. [ 12 ] Let the function f : Z → Z take only integer inputs and have integer outputs. For any integers xand y , f satisfiesf ( x ) + f ( y ) = f ( x + 1) + f ( y − 1)
    If f (2016) = 6102 and f (6102) = 2016, what is f (1)?
    Proposed by: Henrik Boecken
    Answer: 8117
    We havef ( x + 1) = f ( x ) + f ( y ) − f ( y − 1)
    If y is fixed, we havef ( x + 1) = f ( x ) + constantimplying f is linear. Using our two points, then, we get f ( x ) = 8118 − x , so f (1) = 8117