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Supported by British Mathematical OlympiadRound 2 : Tuesday, 30 January 2007 Time allowed Three and a half hours. Each question is worth 10 marks. Instructions •Ful l written solutions - not just answers - are required, with complete proofs of any assertions you may make. Marks awarded wil l depend on the clarity of your mathematical presentation. Work in rough first, and then draft your final version careful ly before writing up your best attempt. Rough work shouldbe handed in, but should be clearly marked. • One or two completesolutions wil l gain far more credit than partial attempts at al l four problems. • The use of rulers and compasses is al lowed, but calculators and protractors are forbidden. • Staple al l the pages neatly together in the top left hand corner, with questions 1,2,3,4 in order, and the cover sheet at the front. In early March, twenty students will be invited to attend the training session to be held at Trinity College, Cambridge (29th March - 2nd April). At the training session, students sit a pair of IMO-style papers and 8 students will be selected for further training. Those selected will be expected to participate in correspondence work and to attend further training. The UK Team of six for this summer’s International Mathematical Olympiad (to be held in Hanoi, Vietnam 23-31 July) will then be chosen. Do not turn over until told to do so. Supported by 2006/7 British Mathematical Olympiad Round 2 1. Triangle AB Chas integer-length sides, and AC= 2007. The internal bisector of 6 B AC meets B CatD. Given that AB=C D , determine AB and B C. 2. Show that there are infinitely many pairs of positive integers ( m, n) such that m+ 1 n + n + 1 m is a positive integer. 3. Let AB Cbe an acute-angled triangle with AB > ACand6 B AC = 60 o . Denote the circumcentre by Oand the orthocentre by Hand let OH meet ABatPand ACatQ. Prove that P O=H Q . Note: The circumcentre of triangle AB Cis the centre of the circle which passes through the vertices A, BandC. The orthocentre is the point of intersection of the perpendiculars from each vertex to the o pposite side. 4. In the land of Hexagonia, the six cities are connected by a rail network such that there is a direct rail line connecting each pair of cities. On Sundays, some lines may be closed for repair. The passengers’ rail charter stipulates that any city must be accessible by rail from any other (not necessarily directly) at all times. In how many different ways can some of the lines be closed sub ject to this condition?