NL-NWO 1987

For $x >0$ , prove that $$\frac{1}{2\sqrt{x+1}}<\sqrt{x+1}-\sqrt{x}<\frac{1}{2\sqrt{x}}$$ and for all $n \ge 2$ prove that $$1 <2\sqrt{n} - \sum_{k=1}^n\frac{1}{\sqrt{k}}<2$$

There are two kinds of creatures living in the flatland of Pentagonia: the Spires ( $S$ ) and the Bones ( $B$ ). They all have the shape of an isosceles triangle: the Spiers have an apical angle of $36^o$ and the bones an apical angle of $108^o$ . Every year on Great Day of Division (September 11 - the day this Olympiad was held) they divide into pieces: each $S$ into two smaller $S$ 's and a $B$ ; each $B$ in an $S$ and a $B$ . Over the course of the year they then grow back to adult proportions. In the distant past, the population originated from one $B$ -being. Deaths do not occur. Investigate whether the ratio between the number of Spires and the number of Bones will eventually approach a limit value and if so, calculate that limit value.

On each side of a regular tetrahedron with edges of length $1$ one constructs exactly such a tetrahedron. This creates a dodecahedron with $8$ vertices and $18$ edges. We imagine that the dodecahedron is hollow. Calculate the length of the largest line segment that fits entirely within this dodecahedron.