NL-NWO 1983

A triangle $ABC$ can be divided into two isosceles triangles by a line through $A$ . Given that one of the angles of the triangles is $30^{\circ}$ , find all possible values of the other two angles.

Prove that if $n$ is an odd positive integer, then the last two digits of $2^{2n}(2^{2n+1}-1)$ in base $10$ are $28$ .

Suppose that $a,b,c,p$ are real numbers with $a,b,c$ not all equal, such that: $a+\frac{1}{b}=b+\frac{1}{c}=c+\frac{1}{a}=p.$ Determine all possible values of $p$ and prove that $abc+p=0$ .

Within an equilateral triangle of side $15$ are $111$ points. Prove that it is always possible to cover three of these points by a round coin of diameter $\sqrt{3}$ , part of which may lie outside the triangle.