Find all positive integers $m$ and $n$ such that $(2^{2^{n}}+1)(2^{2^{m}}+1) $ is divisible by $m\cdot n $ .

At a mathematical competition $n$ students work on $6$ problems each one with three possible answers. After the competition, the Jury found that for every two students the number of the problems, for which these students have the same answers, is $0$ or $2$. Find the maximum possible value of $n$.

For $a,b,c,d>0$ prove that $$\frac {a+\sqrt{ab}+\sqrt[3]{abc}+\sqrt[4]{abcd}}{4} \leq \sqrt[4]{a.\frac{a+b}{2}.\frac{a+b+c}{3}.\frac{a+b+c+d}{4}}$$

Determine whether there exist a positive integer $n<10^9$, such that $n$ can be expressed as a sum of three squares of positive integers by more than $1000$ distinct ways?

Let $\triangle {ABC} $ be isosceles triangle with $AC=BC$ . The point $D$ lies on the extension of $AC$ beyond $C$ and is that $AC>CD$. The angular bisector of $ \angle BCD $ intersects $BD$ at point $N$ and let $M$ be the midpoint of $BD$. The tangent at $M$ to the circumcircle of triangle $AMD$ intersects the side $BC$ at point $P$. Prove that points $A,P,M$ and $N$ lie on a circle.

Let $n$ be positive integer.A square $A$ of side length $n$ is divided by $n^2$ unit squares. All unit squares are painted in $n$ distinct colors such that each color appears exactly $n$ times. Prove that there exists a positive integer $N$ , such that for any $n>N$ the following is true: There exists a square $B$ of side length $\sqrt{n}$ and side parallel to the sides of $A$ such that $B$ contains completely cells of $4$ distinct colors.