[PDF] UKMT - BMO Round 2 - British Mathematical Olympiad 2016.pdf | Plain Text

United Kingdom Mathematics Trust British Mathematical Olympiad Round 2 : Thursday, 28 January 2016 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 time zones, please do not discuss any aspect of the paper on the internet until 8am GMT on Friday 29 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 (31 March-4 April 2016). 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 Hong Kong, China 6–16 July 2016) will then be chosen. Do not turn over until told to do so. United Kingdom Mathematics Trust 2015/16 British Mathematical Olympiad Round 2 1. Circles of radius r 1, r 2 and r 3 touch each other externally, and they touch a common tangent at points A,B and Crespectively, where B lies between Aand C. Prove that 16( r 1 + r 2 + r 3) ≥ 9(AB +B C +C A ). 2. Alison has compiled a list of 20 hockey teams, ordered by ho w good she thinks they are, but refuses to share it. Benjamin may men tion three teams to her, and she will then choose either to tell him which she thinks is the weakest team of the three, or which she think s is the strongest team of the three. Benjamin may do this as many time s as he likes. Determine the largest Nsuch that Benjamin can guarantee to be able to find a sequence T 1, T 2, . . . , T Nof teams with the property that he knows that Alison thinks that T i is better than T i+1 for each 1 ≤ i < N . 3. Let AB C D be a cyclic quadrilateral. The diagonals ACand B D meet at P, and DAand C Bproduced meet at Q. The midpoint of ABisE. Prove that if P Qis perpendicular to AC, then P Eis perpendicular to B C . 4. Suppose that pis a prime number and that there are different positive integers uand vsuch that p2 is the mean of u2 and v2 . Prove that 2 p − u− vis a square or twice a square.