AT-OEMO-FED1 2006

Let $n$ be a non-negative integer, which ends written in decimal notation on exactly $k$ zeros, but which is bigger than $10^k$ . For a $n$ is only $k=k(n)\geq2$ known. In how many different ways (as a function of $k=k(n)\geq2$ ) can $n$ be written as difference of two squares of non-negative integers at least?

Show that the sequence $a_n = \frac {(n + 1)^nn^{2 - n}}{7n^2 + 1}$ is strictly monotonically increasing, where $n = 0,1,2, \dots$ .

In the triangle $ABC$ let $D$ and $E$ be the boundary points of the incircle with the sides $BC$ and $AC$ . Show that if $AD=BE$ holds, then the triangle is isoceles.

Given is the function $f= \lfloor x^2 \rfloor + \{ x \}$ for all positive reals $x$ . ( $\lfloor x \rfloor$ denotes the largest integer less than or equal $x$ and $\{ x \} = x - \lfloor x \rfloor$ .) Show that there exists an arithmetic sequence of different positive rational numbers, which all have the denominator $3$ , if they are a reduced fraction, and don't lie in the range of the function $f$ .