Find all polynomials $P(x)$ with real coefficients such that \[xP\bigg(\frac{y}{x}\bigg)+yP\bigg(\frac{x}{y}\bigg)=x+y\] for all nonzero real numbers $x$ and $y$.

Find the smallest integer $n$ such that each subset of $\{1,2,\ldots, 2004\}$ with $n$ elements has two distinct elements $a$ and $b$ for which $a^2-b^2$ is a multiple of $2004$.

In a convex hexagon $ABCDEF$, triangles $ACE$ and $BDF$ have the same circumradius $R$. If triangle $ACE$ has inradius $r$, prove that \[ \text{Area}(ABCDEF)\le\frac{R}{r}\cdot\text{Area}(ACE).\]

How many integers $n>1$ are there such that $n$ divides $x^{13}-x$ for every positive integer $x$?

A collection of cardboard circles, each with a diameter of at most $1$, lie on a $5\times 8$ table without overlapping or overhanging the edge of the table. A cardboard circle of diameter $2$ is added to the collection. Prove that this new collection of cardboard circles can be placed on a $7\times 7$ table without overlapping or overhanging the edge.

Consider a partition of $\{1,2,\ldots,900\}$ into $30$ subsets $S_1,S_2,\ldots,S_{30}$ each with $30$ elements. In each $S_k$, we paint the fifth largest number blue. Is it possible that, for $k=1,2,\ldots,30$, the sum of the elements of $S_k$ exceeds the sum of the blue numbers?