Sofia has $5$ pieces of paper on a table. He takes some of the pieces, cuts each one into $5$ little pieces, and puts them back on the table. She repeats this procedure several times until she gets tired. Could Sofia end up with $2010$ pieces on the table?

Of all the fractions $\frac{x}{y}$ that satisfy $$\frac{41}{2010}<\frac{x}{y}<\frac{1}{49}$$find the one with the smallest denominator.

In triangle $ABC$, $\angle BAC= 60^o$. Angle bisector of $\angle ABC$ meets side $AC$ at $X$ and angle bisector of $\angle BCA$ meets side $AB$ at $Y$. Prove that if $I$ is the incenter of triangle $ABC$, then $IX=IY$.

In a group of people, every two of them have exactly one mutual friend in the group. Prove that there is one person who is friends with all the other people in the group. Note: the friendship is mutual, that is, if $X$ is friends with $Y$, then $Y$ is friends with $X$.