SE-SKOLORNAS 2012

The function $f$ satisfies the condition $$f (x + 1) = \frac{1 + f (x)}{1 - f (x)}$$ for all real $x$ , for which the function is defined. Determine $f(2012)$ , if we known that $f(1000)=2012$ .

The number $201212200619$ has a factor $m$ such that $6 \cdot 10^9 <m <6.5 \cdot 10^9$ . Find $m$ .

The catheti $AC$ and $BC$ in a right-angled triangle $ABC$ have lengths $b$ and $a$ , respectively. A circle centered at $C$ is tangent to hypotenuse $AB$ at point $D$ . The tangents to the circle through points $A$ and $B$ intersect the circle at points $E$ and $F$ , respectively (where $E$ and $F$ are both different from $D$ ). Express the length of the segment $EF$ in terms of $a$ and $b$ .

Given that $a$ is a real solution to the polynomial equation $$nx^n-x^{n-1}-x^{n-2}-\cdots-x-1=0$$ where $n$ is a positive integer, show that $a=1$ or $-1<a<0$ .

The vertices of a regular $13$ -gon are colored in three different colors. Show that there are three vertices which have the same color and are also the vertices of an isosceles triangle.

A circle is inscribed in an trapezoid. Show that the diagonals of the trapezoid intersect at a point on the diameter of the circle perpendicular to the two parallel sides.