## [

 [ Systems of linear equationsFind all functions $f(x)$, defined for all real $x$ and satisfying the equation $2 f(x)+f(1$ $-x)=x^{2}$

#
Substitute $1-x$ for $x$.

## Solution

In the given equation, substitute $1-x$ for $x$. We obtain the system

$$
\begin{aligned}
& 2 f(x)+f(1-x)=x^{2} \\
& f(x)+2 f(1-x)=(1-x)^{2}
\end{aligned}
$$

from which we find $f(x)=\frac{1}{3}\left(2 x^{2}-(1-x)^{2}\right)=\frac{1}{3}\left(x^{2}+2 x-1\right)$. Verification shows that the found function satisfies the condition.

## Answer

$f(x)=\frac{1}{3}\left(x^{2}+2 x-1\right)$