[PDF] UKMT - BMO Round 2 - British Mathematical Olympiad 2011.pdf | Plain Text

United Kingdom Mathematics Trust British Mathematical Olympiad Round 2 : Thursday, 27 January 2011 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 on Friday 28 January GMT. 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 (14-18 April 2011). 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 Amsterdam, The Netherlands 16–24 July) will then be chosen. Do not turn over until told to do so. United Kingdom Mathematics Trust 2010/11 British Mathematical Olympiad Round 2 1. Let AB Cbe a triangle and Xbe a point inside the triangle. The lines AX, B X andC X meet the circle AB Cagain at P, QandR respectively. Choose a point Uon X P which is between Xand P. Suppose that the lines through Uwhich are parallel to ABand C A meet X QandX R at points Vand Wrespectively. Prove that the points R, W, V andQlie on a circle. 2. Find all positive integers xand ysuch that x+ y+ 1 divides 2 xyand x + y− 1 divides x2 + y2 − 1. 3. The function fis defined on the positive integers as follows; f (1) = 1; f (2 n) = f(n ) if nis even; f (2 n) = 2 f(n ) if nis odd; f (2 n+ 1) = 2 f(n ) + 1 if nis even; f (2 n+ 1) = f(n ) if nis odd. Find the number of positive integers nwhich are less than 2011 and have the property that f(n ) = f(2011). 4. Let Gbe the set of points ( x, y) in the plane such that xand yare integers in the range 1 ≤x, y ≤2011. A subset Sof Gis said to be paral lelogram-free if there is no proper parallelogram with all its vertices in S. Determine the largest possible size of a parallelogram- free subset of G. Note that a proper paral lelogram is one where its vertices do not al l lie on the same line