IMO 2002

The circle $S$ has centre $O$ , and $BC$ is a diameter of $S$ . Let $A$ be a point of $S$ such that $\angle AOB<120{{}^\circ}$ . Let $D$ be the midpoint of the arc $AB$ which does not contain $C$ . The line through $O$ parallel to $DA$ meets the line $AC$ at $I$ . The perpendicular bisector of $OA$ meets $S$ at $E$ and at $F$ . Prove that $I$ is the incentre of the triangle $CEF.$ Attachments:

Find all pairs of positive integers $m,n\geq3$ for which there exist infinitely many positive integers $a$ such that $$\frac{a^m+a-1}{a^n+a^2-1}$$ is itself an integer.

Laurentiu Panaitopol, Romania

Attachments:

Find all functions $f$ from the reals to the reals such that $$\left(f(x)+f(z)\right)\left(f(y)+f(t)\right)=f(xy-zt)+f(xt+yz)$$ for all real $x,y,z,t$ .