Prove that $\frac{1}{1\times2013}+\frac{1}{2\times2012}+\frac{1}{3\times2011}+...+\frac{1}{2012\times2}+\frac{1}{2013\times1}<1$

Point $M$ is an arbitrary point in the plane and let points $G$ and $H$ be the intersection points of the tangents from point M and the circle $k$. Let $O$ be the center of the circle $k$ and let $K$ be the orthocenter of the triangle $MGH$. Prove that ${\angle}GMH={\angle}OGK$.

Find all positive integers $n$ which are divisible by 11 and satisfy the following condition: all the numbers which are generated by an arbitrary rearrangement of the digits of $n$, are also divisible by 11.

In a convex quadrilateral $ABCD$, $E$ is the intersection of $AB$ and $CD$, $F$ is the intersection of $AD$ and $BC$ and $G$ is the intersection of $AC$ and $EF$. Prove that the following two claims are equivalent: $(i)$ $BD$ and $EF$ are parallel. $(ii)$ $G$ is the midpoint of $EF$.

Prove that there exist infinitely many pairwisely disjoint sets $A(1), A(2),...,A(2014)$ which are not empty, whose union is the set of positive integers and which satisfy the following condition: For arbitrary positive integers $a$ and $b$, at least two of the numbers $a$, $b$ and $GCD(a,b)$ belong to one of the sets $A(1), A(2),...,A(2014)$.