Time Allowed: Three and a half hours

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. The line PN meets the circle C₂ again at R. Prove that MQ bisects angle PMR.

2 Show that, for every positive integer n,

121ⁿ - 25ⁿ + 1900ⁿ - (-4)ⁿ

is divisible by 2000.

 

3 Triangle ABC has a right angle at A. Among all the points P on the perimeter of the triangle, find the position of P such that

AP + BP + CP

is minimised

 

4 For each positive integer k, define the sequence {an} by

a₀ = 1 and a_n = kn + (-1)ⁿa_n-1 for each n > or = 1

Determine all values of k for which 2000 is a term of the sequence.

 

5 The seven dwarves decide to form four teams to compete in the Millenium Quiz. Of course, the sizes of the teams will not all be equal. For instance, one team might consist of Doc alone, one of Dopey alone, one of Sleepy, Happy and Grumpy as a trio, and one of Bashful and Sneezy as a pair. In how many ways can the four teams be made up? (The order of the teams or of the dwarves withing the teams does not matter, but each dwarf must be in exactlt one of the teams.)

Suppose Snow White agreed to take part as well. In how many ways could the four teams then be formed?