International Mathematical Olympiad 2002 — Official contest Question 5

Find all functions $f \colon \mathbb{R} \to \mathbb{R}$ such that $$
\left(f(x)+f(z)\right)\left(f(y)+f(t)\right)
= f(xy-zt)+f(xt+yz)
$$ for all real numbers $x$, $y$, $z$, $t$.