## Problem 3

Find all functions $f: \Re \rightarrow \Re$ such that

$$
x^{2} \cdot f(x)+f(1-x)=2 x-x^{4}
$$
Solution:

Substitute $x$ with $1-x$ and we get

$$
(1-x)^{2} \cdot f(1-x)+f(x)=2(1-x)-(1-x)^{4}
$$

By eliminating $f(1-x)$ between this equation and the given one, we obtain $f(x)=1-x^{2}$.

It is verified that $f(x)=1-x^{2}$ satisfies the proposed equation.

Therefore, the solution sought is $f(x)=1-x^{2}$.