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Supported by British Mathematical OlympiadRound 2 : Tuesday, 31 January 2006 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 (6-10 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 6 for this summer’s International Mathematical Olympiad (to be held in Ljubljana, Slovenia 10-18 July) will then be chosen. Do not turn over until told to do so. Supported by 2005/6 British Mathematical Olympiad Round 2 1. Find the minimum possible value of x2 + y2 given that xand yare real numbers satisfying xy(x 2 − y2 ) = x2 + y2 and x6 = 0 . 2. Let xand ybe positive integers with no prime factors larger than 5. Find all such xand ywhich satisfy x2 − y2 = 2 k for some non-negative integer k. 3. Let AB Cbe a triangle with AC > AB. The point Xlies on the side B Aextended through A, and the point Ylies on the side C Ain such a way that B X=C A and C Y=B A . The line X Ymeets the perpendicular bisector of side B CatP. Show that 6 B P C +6 B AC = 180o . 4. An exam consisting of six questions is sat by 2006 children. Each question is marked either right or wrong. Any three children have right answers to at least five of the six questions between them. Let N be the total number of right answers achieved by all the children (i.e. the total number of questions solved by child 1 + the total solved by child 2 + + the total solved by child 2006). Find the least possible value of N.