SILK-ROAD 2012

Trapezium $ABCD$ , where $BC||AD$ , is inscribed in a circle, $E$ is midpoint of the arc $AD$ of this circle not containing point $C$ . Let $F$ be the foot of the perpendicular drawn from $E$ on the line tangent to the circle at the point $C$ . Prove that $BC=2CF$ .

In each cell of the table $4 \times 4$ , in which the lines are labeled with numbers $1,2,3,4$ , and columns with letters $a,b,c,d$ , one number is written: $0$ or $1$ . Such a table is called valid if there are exactly two units in each of its rows and in each column. Determine the number of valid tables.

Let $n > 1$ be an integer. Determine the greatest common divisor of the set of numbers $\left\{ \left( \begin{matrix} 2n 2i+1 \end{matrix} \right):0 \le i \le n-1 \right\}$ i.e. the largest positive integer, dividing $\left( \begin{matrix} 2n 2i+1 \end{matrix} \right)$ without remainder for every $i = 0, 1, ..., n-1$ . (Here $\left( \begin{matrix} m l \end{matrix} \right)=\text{C}_{m}^{l}=\frac{m\text{!}}{l\text{!}\left( m-l \right)\text{!}}$ is binomial coefficient.)

Prove that for any positive integer $n$ , the arithmetic mean of $\sqrt[1]{1},\sqrt[2]{2},\sqrt[3]{3},\ldots ,\sqrt[n]{n}$ lies in $\left[ 1,1+\frac{2\sqrt{2}}{\sqrt{n}} \right]$ .