HMMT 十一月 2022 · 团队赛 · 第 1 题

HMMT November 2022 — Team Round — Problem 1

[20] Two linear functions f ( x ) and g ( x ) satisfy the properties that for all x , • f ( x ) + g ( x ) = 2 • f ( f ( x )) = g ( g ( x )) and f (0) = 2022. Compute f (1).

解析

[20] Two linear functions f ( x ) and g ( x ) satisfy the properties that for all x , • f ( x ) + g ( x ) = 2 • f ( f ( x )) = g ( g ( x )) and f (0) = 2022. Compute f (1). Proposed by: Vidur Jasuja Answer: 1 Solution 1: Firstly, f ( x ) and g ( x ) must intersect – otherwise, f ( x ) = g ( x ) = 1, which can’t be true. Secondly, suppose they intersect at a , so that f ( a ) = g ( a ) = c . Then f ( c ) = g ( c ) = ⇒ f ( c ) = g ( c ) = 1. But then, a = c , and c = 1. So f (1) = 1, and we’re done. Solution 2: We will solve the problem manually, setting f ( x ) = ax + 2022 and g ( x ) = − ax − 2020. 2 2 Then f ( f ( x )) = a x + 2022 a + 2022, while g ( g ( x )) = a x + 2020 a − 2020 . Then, a = − 2021, so then f (1) = 1 .