DE-MO 1997

Prove that there are no perfect squares $a,b,c$ such that $ab-bc = a$ .

For a positive integer $k$ , let us denote by $u(k)$ the greatest odd divisor of $k$ . Prove that, for each $n \in N$ , $\frac{1}{2^n} \sum_{k = 1}^{2^n} \frac{u(k)}{k}> \frac{2}{3}$ .

In a convex quadrilateral $ABCD$ we are given that $\angle CBD = 10^o$ , $\angle CAD = 20^o$ , $\angle ABD = 40^o$ , $\angle BAC = 50^o$ . Determine the angles $\angle BCD$ and $\angle ADC$ .

Find all real solutions $(x,y,z)$ of the system of equations $$\begin{cases} x^3 = 2y-1 y^3 = 2z-1 z^3 = 2x-1\end{cases}$$

We are given $n$ discs in a plane, possibly overlapping, whose union has the area $1$ . Prove that we can choose some of them which are mutually disjoint and have the total area greater than $1/9$ .