1 For each positive integer n, let S_n denote the set consisting of the first n natural numbers, that is

(i) For which values of n is it possible to express S_n as the union of two non-empty disjoint subsets so that the elements in the two subsets have equal sum?

(ii) For which values of n is it possible to express S_n as the union of three non-empty disjoint subsets so that the elements in the three subsets have equal sum?

2 Let ABCDEF be a hexagon (which may not be regular), which circumscribes a circle S. (That is, S is tangent to each of the six sides of the hexagon.) The circle S touches AB, CD, EF at their midpoints P, Q, R respectively. Let X, Y, Z be the points of contact of S with BC, DE, FA respectively. Prove that PY, QZ, RX are concurrent.

3 Non-negative real numbers p , q and r satisfy p + q + r = 1. Prove that

4 Consider all numbers of the form 3n² + n + 1, where n is a positive integer.

(i) How small can the sum of the digits (in base 10) of such a number be?

(ii) Can such a number have the sum of its digits (in base 10) equal to 1999?