[PDF] UKMT - BMO Round 2 - British Mathematical Olympiad 2001.pdf | Plain Text

BRITISH MATHEMATICAL OLYMPIAD Round 2 : Tuesday, 27 February 2001 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 will be invited to attend the training session to be held at Trinity College, Cambridge (8-11 April). On the final morning of the training session, students sit a paper with just 3 Olympiad-style problems, and 8 students will be selected for further training. Those selected will be expected to participate in correspondence work and to attend another meeting in Cambridge (probably 26-29 May). The UK Team of 6 for this summer’s International Mathematical Olympiad (to be held in Washington DC, USA, 3-14 July) will then be chosen. Do not turn over until told to do so. 2001 British Mathematical Olympiad Round 2 1. Ahmed and Beth have respectively pand qmarbles, with p > q . Starting with Ahmed, each in turn gives to the other as many marbles as the other already possesses. It is found that after 2 n such transfers, Ahmed has qmarbles and Beth has p marbles. Find p qin terms of n. 2. Find all pairs of integers ( x, y) satisfying 1 + x2 y = x2 + 2 xy+ 2 x+ y. 3. A triangle AB Chas6 AC B > 6 AB C . The internal bisector of 6 B AC meets B CatD. The point Eon AB is such that 6 E DB = 90◦ . The point Fon AC is such that 6 B E D =6 DE F . Show that 6 B AD =6 F DC . 4. Ndwarfs of heights 1 ,2 ,3 , . . . , N are arranged in a circle. For each pair of neighbouring dwarfs the positive difference between the heights is calculated; the sum of these N differences is called the “total variance” Vof the arrangement. Find (with proof ) the maximum and minimum possible values of V.