3. Let the function $f(x, y)$ satisfy:
(1) $f(x, x)=x$;
(2) $f(k x, k y)=k f(x, y)$;
(3) $f\left(x_{1}+x_{2}, y_{1}+y_{2}\right)=f\left(x_{1}, y_{1}\right)+f\left(x_{2}, y_{2}\right)$;
(4) $f(x, y)=f\left(y, \frac{x+y}{2}\right)$.

Then $f(x, y)=$ . $\qquad$
3. $\frac{x}{3}+\frac{2 y}{3}$.

Notice that,
$$
\begin{array}{l}
f(x, 0)+f(0, x) \stackrel{(3)}{=} f(x, x)=x, \\
f(x, 0) \stackrel{(4)}{=} f\left(0, \frac{x}{2}\right) \stackrel{(2)}{=} \frac{1}{2} f(0, x) .
\end{array}
$$

Subtracting the two equations gives $f(0, x)=\frac{2 x}{3}$.
Similarly, $f(0, y)=\frac{2 y}{3}$.
Thus, $f(x, 0)=\frac{x}{3}$.
Therefore, $f(x, y) \stackrel{(3)}{=} f(x, 0)+f(0, y)=\frac{x}{3}+\frac{2 y}{3}$.