AT-OEMO-REG 2005

Show for all integers $n \ge 2005$ the following chaine of inequalities: $(n+830)^{2005}<n(n+1)\dots(n+2004)<(n+1002)^{2005}$

Construct the semicircle $h$ with the diameter $AB$ and the midpoint $M$ . Now construct the semicircle $k$ with the diameter $MB$ on the same side as $h$ . Let $X$ and $Y$ be points on $k$ , such that the arc $BX$ is $\frac{3}{2}$ times the arc $BY$ . The line $MY$ intersects the line $BX$ in $D$ and the semicircle $h$ in $C$ . Show that $Y$ ist he midpoint of $CD$ .

For which values of $k$ and $d$ has the system $x^3+y^3=2$ and $y=kx+d$ no real solutions $(x,y)$ ?

Prove: if an infinte arithmetic sequence ( $a_n=a_0+nd$ ) of positive real numbers contains two different powers of an integer $a>1$ , then the sequence contains an infinite geometric sequence ( $b_n=b_0q^n$ ) of real numbers.