Frog is in the origin of decartes coordinate system. Every second frog jumpes horizontally or vertically in some of the $4$ adjacent points which coordinates are integers. Find number of different points in which frog can be found in $2015$ seconds.

Two different $3$ digit numbers are picked and then for every of them is calculated sum of all $5$ numbers which are getting when digits of picked number change place (etc. if one of the number is $707$, the sum is $2401=770+77+77+770+707$). Do the given results must be different?

Prove inequallity : $$1+\frac{1}{2^3}+...+\frac{1}{2015^3}<\frac{5}{4}$$

The diagonals $AD$, $BE$, $CF$ of cyclic hexagon $ABCDEF$ intersect in $S$ and $AB$ is parallel to $CF$ and lines $DE$ and $CF$ intersect each other in $M$. Let $N$ be a point such that $M$ is the midpoint of $SN$. Prove that circumcircle of $\triangle ADN$ is passing through midpoint of segment $CF$.