DK-GEORGMOHR 1991

Describe the amount of points $P(x, y)$ that are twice as far apart $A(3, 0)$ as to $0(0, 0)$ .

Prove that for $0<x<\frac{\pi}{2}$ , $$\sin x + \tan x > 2x$$

A right-angled triangle has perimeter $60$ and the altitude of the hypotenuse has a length $12$ . Determine the lengths of the sides.

Let $a, b, c$ and $d$ be arbitrary real numbers. Prove that if $$a^2+b^2+c^2+d^2=ab+bc+cd+da,$$ then $a=b=c=d$ .

Show that no matter how $15$ points are plotted within a circle of radius $2$ (circle border included), there will be a circle with radius $1$ (circle border including) which contains at least three of the $15$ points.