DK-GEORGMOHR 2010

Four right triangles, each with the sides $1$ and $2$ , are assembled to a figure as shown. How large a fraction does the area of the small circle make up of that of the big one?

Prove that for any integer $n$ there exist integers $a, b$ and $c$ such that $n=a^2+b^2-c^2$ .

Can $29$ boys and $31$ girls be lined up holding hands so no one is holding hands with two girls?

An equilateral triangle $ABC$ is given. With $BC$ as diameter, a semicircle is drawn outside the triangle. On the semicircle, points $D$ and $E$ are chosen such that the arc lengths $BD, DE$ and $EC$ are equal. Prove that the line segments $AD$ and $AE$ divide the side $BC$ into three equal parts.