Find all prime numbers $p,q$ for which $pq$ divides $(5^p-2^p)(5^q-2^q)$.

Find the side length of the smallest equilateral triangle in which three discs of radii $2,3,4$ can be placed without overlap.

The quadratic polynomials $f$ and $g$ with real coefficients are such that if $g(x)$ is an integer for some $x>0$, then so is $f(x)$. Prove that there exist integers $m,n$ such that $f(x)=mg(x)+n$ for all $x$.

Sequence $\{a_n\}$ it define $a_1=1$ and \[a_{n+1}=\frac{a_n}{n}+\frac{n}{a_n}\]for all $n\ge 1$ Prove that $\lfloor a_n^2\rfloor=n$ for all $n\ge 4.$

The quadrilateral $ABCD$ is inscribed in a circle. The lines $AB$ and $CD$ meet each other in the point $E$, while the diagonals $AC$ and $BD$ in the point $F$. The circumcircles of the triangles $AFD$ and $BFC$ have a second common point, which is denoted by $H$. Prove that $\angle EHF=90^\circ$.

A square table of size $7\times 7$ with the four corner squares deleted is given. What is the smallest number of squares which need to be colored black so that a $5-$square entirely uncolored Greek cross (Figure 1) cannot be found on the table? Prove that it is possible to write integers in each square in a way that the sum of the integers in each Greek cross is negative while the sum of all integers in the square table is positive. [asy][asy] size(3.5cm); usepackage("amsmath"); MP("\text{Figure }1.", (1.5, 3.5), N); DPA(box((0,1),(3,2))^^box((1,0),(2,3)), black); [/asy][/asy]