[PDF] UKMT - BMO Round 2 - British Mathematical Olympiad 2009.pdf | Plain Text

United Kingdom Mathematics Trust British Mathematical Olympiad Round 2 : Thursday, 29 January 2009 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 (2-6 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 Bremen, Germany 13-22 July) will then be chosen. Do not turn over until told to do so. United Kingdom Mathematics Trust 2008/9 British Mathematical Olympiad Round 2 1. Find all solutions in non-negative integers a, bto√ a + √ b = √ 2009. 2. Let AB Cbe an acute-angled triangle with 6 B =6 C . Let the circumcentre be Oand the orthocentre be H. Prove that the centre of the circle B OHlies on the line AB.The circumcentre of a triangle is the centre of its circumcircle. The orthocentre of a trian gle is the point where its three altitudes meet. 3. Find all functions ffrom the real numbers to the real numbers which satisfy f(x 3 ) + f(y 3 ) = ( x+ y)( f(x 2 ) + f(y 2 ) − f(xy )) for all real numbers xand y. 4. Given a positive integer n, let b(n ) denote the number of positive integers whose binary representations occur as blocks of consecutive integers in the binary expansion of n. For example b(13) = 6 because 13 = 1101 2, which contains as consecutive blocks the binary representations of 13 = 1101 2, 6 = 110 2, 5 = 101 2, 3 = 11 2, 2 = 10 2 and 1 = 1 2. Show that if n≤ 2500, then b(n ) ≤ 39, and determine the values of nfor which equality holds.