KG-MO 2012

Prove that $n$ must be prime in order to have only one solution to the equation $\frac{1}{x}-\frac{1}{y}=\frac{1}{n}$ , $x,y\in\mathbb{N}$ .

Given positive real numbers ${a_1},{a_2},...,{a_n}$ with ${a_1}+{a_2}+...+{a_n}= 1$ . Prove that $\left({\frac{1}{{a_1^2}}-1}\right)\left({\frac{1}{{a_2^2}}-1}\right)...\left({\frac{1}{{a_n^2}}-1}\right)\geqslant{({n^2}-1)^n}$ .

Prove that if the diagonals of a convex quadrilateral are perpendicular, then the feet of perpendiculars dropped from the intersection point of diagonals on the sides of this quadrilateral lie on one circle. Is the converse true?

Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $f(f(x)^2+f(y)) = xf(x)+y$ , $\forall x,y\in R$ .

The sequence of natural numbers is defined as follows: for any $k\geq 1$ , $a_{k+2}= a_{k+1}\cdot a_k+1$ . Prove that for $k\geq 9$ the number $a_k-22$ is composite.

The numbers $1, 2,\ldots, 50$ are written on a blackboard. Each minute any two numbers are erased and their positive difference is written instead. At the end one number remains. Which values can take this number?