PMO 2008

Prove that the set $\{1, 2, \cdots, 2007\}$ can be expressed as the union of disjoint subsets $A_i$ for $i=1,2,\cdots, 223$ such that each $A_i$ contains nine elements and the sum of all the elements in each $A_i$ is the same.

Find the largest integer $n$ for which $\frac{n^{2007}+n^{2006}+\cdots+n^2+n+1}{n+2007}$ is an integer.

Let $P$ be a point outside a circle $\Gamma$ , and let the two tangent lines through $P$ touch $\Gamma$ at $A$ and $B$ . Let $C$ be on the minor arc $AB$ , and let ray $PC$ intersect $\Gamma$ again at $D$ . Let $\ell$ be the line through $B$ and parallel to $PA$ . $\ell$ intersects $AC$ and $AD$ at $E$ and $F$ , respectively. Prove that $B$ is the midpoint of $EF$ .

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a function defined by $f(x)=\frac{2008^{2x}}{2008+2008^{2x}}$ . Prove that $$\begin{aligned} f\left(\frac{1}{2007}\right)+f\left(\frac{2}{2007}\right)+\cdots+f\left(\frac{2005}{2007}\right)+f\left(\frac{2006}{2007}\right)=1003. \end{aligned}$$