DE-BUNDESWETTBEWERB 2006

Find two consecutive integers with the property that the sums of their digits are each divisible by $2006$ .

Prove that there are no integers $x,y$ for that it is $x^3+y^3=4\cdot(x^2y+xy^2+1)$ .

Let $a,b,c$ be the sidelengths of a triangle such that $a^2+b^2 > 5c^2$ holds. Prove that $c$ is the shortest side of the triangle.

Find all functions $f: Q^{+}\rightarrow R$ such that

$f(x)+f(y)+2xyf(xy)=\frac{f(xy)}{f(x+y)}$ for all $x,y\in Q^{+}$

A point $P$ is given inside an acute-angled triangle $ABC$ . Let $A',B',C'$ be the orthogonal projections of $P$ on sides $BC, CA, AB$ respectively. Determine the locus of points $P$ for which $\angle BAC = \angle B'A'C'$ and $\angle CBA = \angle C'B'A'$

A positive integer is called digit-reduced if at most nine different digits occur in its decimal representation (leading $0$ s are omitted.) Let $M$ be a finite set of digit-reduced numbers. Show that the sum of the reciprocals of the elements in $M$ is less than $180$ .