CZ-SK-MO 2003

Solve the following system in the set of real numbers: $x^2 -xy+y^2 = 7$ , $x^2y+xy^2 = -2$ .

On sides $BC,CA,AB$ of a triangle $ABC$ points $D,E,F$ respectively are chosen so that $AD,BE,CF$ have a common point, say $G$ . Suppose that one can inscribe circles in the quadrilaterals $AFGE,BDGF,CEGD$ so that each two of them have a common point. Prove that triangle $ABC$ is equilateral.

A sequence $(x_n)_{n= 1}^{\infty}$ satisfies $x_1 = 1$ and for each $n > 1, x_n = \pm (n-1)x_{n-1} \pm (n-2)x_{n-2} \pm ... \pm 2x_2 \pm x_1$ . Prove that the signs " $\pm$ " can be chosen so that $x_n \ne 12$ holds only for finitely many $n$ .

Let be given an obtuse angle $AKS$ in the plane. Construct a triangle $ABC$ such that $S$ is the midpoint of $BC$ and $K$ is the intersection point of $BC$ with the bisector of $\angle BAC$ .

Show that, for each integer $z \ge 3$ , there exist two two-digit numbers $A$ and $B$ in base $z$ , one equal to the other one read in reverse order, such that the equation $x^2 -Ax+B$ has one double root. Prove that this pair is unique for a given $z$ . For instance, in base $10$ these numbers are $A = 18, B = 81$ .