AT-OEMO-FED1 2013

Show that if for non-negative integers $m$ , $n$ , $N$ , $k$ the equation $$(n^2+1)^{2^k}\cdot(44n^3+11n^2+10n+2)=N^m$$ holds, then $m = 1$ .

Solve the following system of equations in rational numbers: $$(x^2+1)^3=y+1, (y^2+1)^3=z+1, (z^2+1)^3=x+1.$$

Arrange the positive integers into two lines as follows: \begin{align*} 1 3 6 11 \ 19 \qquad32 53\ldots \text{\ \ } 2 4\ \ 5 7\ \ 8\ \ 9\ \ 10 \ 12\ 13\ 14\ 15\ 16\ 17\ 18 \ 20 \text{ to } 31 \ 33 \text{ to } 52 \ \ldots\end{align*} We start with writing $1$ in the upper line, $2$ in the lower line and $3$ again in the upper line. Afterwards, we alternately write one single integer in the upper line and a block of integers in the lower line. The number of consecutive integers in a block is determined by the first number in the previous block. Let $a_1$ , $a_2$ , $a_3$ , $\ldots$ be the numbers in the upper line. Give an explicit formula for $a_n$ .

Let $A$ , $B$ and $C$ be three points on a line (in this order). For each circle $k$ through the points $B$ and $C$ , let $D$ be one point of intersection of the perpendicular bisector of $BC$ with the circle $k$ . Further, let $E$ be the second point of intersection of the line $AD$ with $k$ . Show that for each circle $k$ , the ratio of lengths $\overline{BE}:\overline{CE}$ is the same.