In a tennis tournament there are 2n participants. In the first round of the tournament each participant plays just once, so there are n games, each occupying a pair of players. Show that the pairing for the first round can be arranged in exactly

1x3x5X7X9X.......X(2n-1)

different ways.

2. In a tetrahedron (which is not necessarily regular) two opposite edges have the same length a and they are perpendicular to each other. Moreover they are each perpendicular to a line of length b which joins their midpoints. Express the volume of the tetrahedron in terms of a and b, and prove your answer.

3. Consider the following four propositions, which are not necessarily true.

I. If a polygon inscribed in a circle is equilateral it is also equiangular.

II. If a polygon inscribed in a circle is equiangular it is also equilateral.

III. If a polygon circumscribed about a circle is equilateral it is also equiangular.

Iv. If a polygon circumscribed about a circle is equiangular it is also equilateral.

(A) State which of the four propositions are true and which are false, giving a proof of your statement in each case.

(B) If, instead of general polygons, we should consider only quadrilaterals which of the four propositions are true and which are false? And if we consider only pentagons? In answering (B) you may state conjectures, but prove as much as you can and separate clearly what is proved and what is not.