ÖMO 1990

Determine the number of integers $n$ with $1 \le n \le N=1990^{1990}$ such that $n^2-1$ and $N$ are coprime.

Show that for all integers $n \ge 2$ , $\sqrt { 2\sqrt[3]{3 \sqrt[4]{4...\sqrt[n]{n}}}}<2$

In a convex quadrilateral $ABCD$ , let $E$ be the intersection point of the diagonals, and let $F_1,F_2,$ and $F$ be the areas of $ABE,CDE,$ and $ABCD,$ respectively. Prove that:

$\sqrt {F_1}+\sqrt {F_2} \le \sqrt {F}.$

For each nonzero integer $n$ find all functions $f: \mathbb{R} - \{-3,0 \} \rightarrow \mathbb{R}$ satisfying:

$f(x+3)+f \left( -\frac{9}{x} \right)=\frac{(1-n)(x^2+3x-9)}{9n(x+3)}+\frac{2}{n}$ for all $x \not= 0,-3.$

Furthermore, for each fixed $n$ find all integers $x$ for which $f(x)$ is an integer.

Determine all rational numbers $r$ such that all solutions of the equation:

$rx^2+(r+1)x+(r-1)=0$ are integers.

A convex pentagon $ABCDE$ is inscribed in a circle. The distances of $A$ from the lines $BC,CD,DE$ are $a,b,c,$ respectively. Compute the distance of $A$ from the line $BE$ .