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British Mathematical Olympiad Round 2 : Tuesday, 26 February 2002 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 (4 – 7 April). On the final morning of the training session, students sit a paper with just 3 Olympiad-style problems, and 8 students will be selected for further training. Those selected will be expected to participate in correspondence work and to attend another meeting in Cambridge. The UK Team of 6 for this summer’s International Mathematical Olympiad (to be held in Glasgow, 22 –31 July) will then be chosen. Do not turn over until told to do so. 2002 British Mathematical Olympiad Round 2 1. The altitude from one of the vertices of an acute-angled triangle AB Cmeets the opposite side at D. From D perpendiculars DEandDF are drawn to the other two sides. Prove that the length of E Fis the same whichever vertex is chosen. 2. A conference hall has a round table wth nchairs. There are n delegates to the conference. The first delegate chooses his or her seat arbitrarily. Thereafter the ( k+ 1) th delegate sits k places to the right of the kth delegate, for 1 ≤k≤ n− 1. (In particular, the second delegate sits next to the first.) No chair can be occupied by more than one delegate. Find the set of values nfor which this is possible. 3. Prove that the sequence defined by y0 = 1 , y n+1 =1 2¡ 3 y n + p 5 y 2 n − 4¢ , (n ≥ 0) consists only of integers. 4. Suppose that B 1, . . . , B Nare Nspheres of unit radius arranged in space so that each sphere touches exactly two others externally. Let Pbe a point outside all these spheres, and let the Npoints of contact be C 1, . . . , C N. The length of the tangent from Pto the sphere B i(1 ≤i≤ N) is denoted by t i. Prove the product of the quantities t i is not more than the product of the distances P C i.