
Problem 3. Find all functions $f(x)$ with integer values and defined in the set of the integers, such that

$$
3 f(x)-2 f(f(x))=x
$$

for all integers $x$.


Solution. The function $f(x)=x$ satisfies the condition of the problem.

Let $f(x)$ be a function which satisfies the condition. Let $g(x)=f(x)-x$. The condition can be written in the form

$$
2 f(f(x))-2 f(x)=f(x)-x
$$

which is equivalent to

$$
g(x)=2 g(f(x))
$$

From here we obtain

$$
g(x)=2 g(f(x))=2^{2} g(f(f(x)))=2^{3} g(f(f(f(x))))=2^{4} g(f(f(f(f(x)))))=\ldots
$$

Since the numbers $g(f(f \ldots f(x)) \ldots)$ are integer, then $g(x)$ is divisible by $2^{n}$ for all integers $x$ and all natural numbers $n$. This is possible only if $g(x)=0$. Thus, $f(x)=x$ is the only solution of the problem.

## Winter mathematics competition-Burgas, 1997
