[PDF] UKMT - BMO Round 2 - British Mathematical Olympiad 2013.pdf | Plain Text

United Kingdom Mathematics Trust British Mathematical Olympiad Round 2 : Thursday, 31 January 2013 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 1 February. 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 (4–8 April 2013). At the training session, students sit a pair of IMO-style papers and eight students will be selected for further training and selection examinations. The UK Team of six for this summer’s IMO (to be held in Santa Marta, Colombia, 18–28 July 2013) will then be chosen. Do not turn over until told to do so. United Kingdom Mathematics Trust 2012/13 British Mathematical Olympiad Round 2 1. Are there infinitely many pairs of positive integers ( m, n) such that both mdivides n2 + 1 and ndivides m2 + 1? 2. The point Plies inside triangle AB Cso that 6 AB P =6 P C A . The point Qis such that P B QCis a parallelogram. Prove that 6 QAB = 6 C AP . 3. Consider the set of positive integers which, when written in binary, have exactly 2013 digits and more 0s than 1s. Let nbe the number of such integers and let sbe their sum. Prove that, when written in binary, n+ shas more 0s than 1s. 4. Suppose that AB C Dis a square and that Pis a point which is on the circle inscribed in the square. Determine whether or not it i s possible that P A,P B ,P C ,P D andABare all integers.