Track / Stage 5 / 61 of 400 #661 of 1964

Problem 661

AIME late

Algebra Difficulty 5.1 Prove it Thai Mathematical Olympiad · Thailand

Does there exist a function $f: \mathbb{Z} \to \mathbb{Z}$ such that
$f(f(n) - 2n) = 2f(n) + n$
for all integers $n$ ?

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

There are many functions $f$ that satisfy the given condition. One of them is given by the following definition:
$f(n) = \begin{cases} n & \text{if } n \ge 0 \\ -3n & \text{if } n < 0 \end{cases}$
which can be verified in the following 3 cases.

$n > 0$ : $f(f(n) - 2n) = f(-n) = 3n$ and $2f(n) + n = 2n + n = 3n$ .

$n < 0$ : $f(f(n) - 2n) = f(-5n) = -5n$ and $2f(n) + n = -5n$ .

$n = 0$ : We have $f(0) = 0$ and $f(f(n) - 2n) = 2f(n) + n = 0$ .

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.