[PDF] UKMT - BMO Round 2 - British Mathematical Olympiad 2010.pdf | Plain Text

United Kingdom Mathematics Trust British Mathematical Olympiad Round 2 : Thursday, 28 January 2010 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 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 (8-12 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 IMO (to be held in Astana, Kazakhstan 6-12 July) will then be chosen. Do not turn over until told to do so. United Kingdom Mathematics Trust 2009/10 British Mathematical Olympiad Round 2 1. There are 2010 2010 children at a mathematics camp. Each has at most three friends at the camp, and if Ais friends with B, then Bis friends with A. The camp leader would like to line the children up so that there are at most 2010 children between any pair of friends. Is it always possible to do this? 2. In triangle AB Cthe centroid is Gand Dis the midpoint of C A. The line through Gparallel to B Cmeets ABatE. Prove that 6 AE C = 6 DGC if, and only if, 6 AC B = 90◦ . The centroid of a triangle is the intersection of the three medians, the lines which join each vertex to the midpoint of the opposite side. 3. The integer xis at least 3 and n= x6 − 1. Let pbe a prime and kbe a positive integer such that pk is a factor of n. Show that p3 k < 8n . 4. Prove that, for all positive real numbers x, yand z, 4( x+ y+ z)3 > 27( x2 y + y2 z + z2 x ).