Time Allowed: Three and a half hours

Each question is worth 10 marks

The use of rulers and compasses is allowed, but calculators and protractors are forbidden

1 Two intersecting circles C₁ and C₂ have a common tangent which touches C₁ at P and C₂ at Q. The two circles intersect at M and N, where N is nearer to PQ than M is. Prove that the triangles MNP and MNQ have equal areas.

2 Given that x, y and z are positive real numbers satisfying xyz=32, find the minimum value of

x² + 4xy + 4y² + 2z²

 

3 Find positive integers a and b such that:

(cuberoota + cuberootb - 1)² = 49 + 20cuberoot6

 

4 (a) Find a set A of ten positive integers such that no six distinct elements of A have a sum which is divisible by 6.

(b) Is it possible to find such a set if "ten" is replaced by "eleven"?