SMO 2011 Senior Round 2

In the triangle $ABC$, the altitude at $A$, the bisector of $\angle B$ and the median at $C$ meet at a common point. Prove that the triangle $ABC$ is equilateral.

Determine if there is a set $S$ of $2011$ positive integers so that for every pair $m,n$ of distinct elements of $S$, $|m-n|=(m,n)$. Here $(m,n)$ denotes the greatest common divisor of $m$ and $n$.

Find all positive integers $n$ such that $$\cos\frac\pi n\cos\frac{2\pi}n\cos\frac{3\pi}n=\frac1{n+1}.$$

Let $n$ and $k$ be positive integers with $n\ge k\ge2$. For $i=1,\ldots,n$, let $S_i$ be a nonempty set of consecutive integers such that among any $k$ of them, there are two with nonempty intersection. Prove that there is a set $X$ of $k-1$ integers such that each $S_i$, $i=1,\ldots,n$, contains at least one integer in $X$.

Given $x_1,x_2,\ldots,x_n>0$, $n\ge5$, show that
$$\frac{x_1x_2}{x_1^2+x_2^2+2x_3x_4}+\frac{x_2x_3}{x_2^2+x_3^2+2x_4x_5}+\cdots+
\frac{x_nx_1}{x_n^2+x_1^2+2x_2x_3}\le\frac{n-1}{2}.$$