HMMT 十一月 2016 · 冲刺赛 · 第 22 题

HMMT November 2016 — Guts Round — Problem 22

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

解析

[ 12 ] Let the function f : Z → Z take only integer inputs and have integer outputs. For any integers x and y , f satisfies f ( 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 have f ( x + 1) = f ( x ) + f ( y ) − f ( y − 1) If y is fixed, we have f ( x + 1) = f ( x ) + constant implying f is linear. Using our two points, then, we get f ( x ) = 8118 − x , so f (1) = 8117