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United Kingdom Mathematics Trust British Mathematical Olympiad Round 2 : Thursday, 26 January 2012 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. • To accommodate candidates sitting in other timezones, please do not discuss any aspect of the paper on the internet until 8am GMT on Friday 27 January. In early March, twenty students eligible to rep- resent the UK at the International Mathematical Olympiad will be invited to attend the training session to be held at Trinity College, Cambridge (29 March – 2 April 2012). At the training session, students sit a pair of IMO-style papers and eight 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 IMO (to be held in Mar del Plata, Argentina, 4–16 July) will then be chosen. Do not turn over until told to do so. United Kingdom Mathematics Trust 2011/12 British Mathematical Olympiad Round 2 1. The diagonals ACand B D of a cyclic quadrilateral meet at E. The midpoints of the sides AB,B C ,C D andDAareP, Q, R andS respectively. Prove that the circles E P SandE QR have the same radius. 2. A function fis defined on the positive integers by f(1) = 1 and, for n > 1, f(n ) = f 2n − 1 3 +f 2n 3 where ⌊x ⌋ denotes the greatest integer less than or equal to x. Is it true that f(n ) − f(n − 1) ≤nfor all n >1? [Here are some examples of the use of ⌊x ⌋ :⌊π ⌋ = 3 ,⌊1729 ⌋= 1729 and ⌊2012 1000 ⌋ = 2 .] 3. The set of real numbers is split into two subsets which do no t intersect. Prove that for each pair ( m, n) of positive integers, there are real numbers x < y < z all in the same subset such that m(z − y) = n(y − x). 4. Show that there is a positive integer kwith the following property: if a, b, c, d, e andfare integers and mis a divisor of a n + bn + cn − dn − en − fn for all integers nin the range 1 ≤n≤ k, then mis a divisor of a n + bn + cn − dn − en − fn for all positive integers n.