CMO 1979

It is known in Euclidean geometry that the sum of the angles of a triangle is constant. Prove, however, that the sum of the dihedral angles of a tetrahedron is not constant.

Let $a$ , $b$ , $c$ , $d$ , $e$ be integers such that $1 \le a < b < c < d < e$ . Prove that $$\frac{1}{[a,b]} + \frac{1}{[b,c]} + \frac{1}{[c,d]} + \frac{1}{[d,e]} \le \frac{15}{16},$$ where $[m,n]$ denotes the least common multiple of $m$ and $n$ (e.g. $[4,6] = 12$ ).

A dog standing at the centre of a circular arena sees a rabbit at the wall. The rabbit runs round the wall and the dog pursues it along a unique path which is determined by running at the same speed and staying on the radial line joining the centre of the arena to the rabbit. Show that the dog overtakes the rabbit just as it reaches a point one-quarter of the way around the arena.