[PDF] UKMT - BMO Round 2 - British Mathematical Olympiad 2004.pdf | Plain Text

Supported by British Mathematical Olympiad Round 2 : Tuesday, 24 February 2004 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 (1-5 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 further training. The UK Team of 6 for this summer’s International Mathematical Olympiad (to be held in Athens, 9-18 July) will then be chosen. Do not turn over until told to do so. Supported by 2004 British Mathematical Olympiad Round 2 1. Let AB Cbe an equilateral triangle and Dan internal point of the side B C. A circle, tangent to B CatD, cuts ABinternally at M and N, and ACinternally at Pand Q. Show that B D+AM +AN =C D +AP +AQ . 2. Show that there is an integer nwith the following properties: (i) the binary expansion of nhas precisely 2004 0s and 2004 1s; (ii) 2004 divides n. 3. (a) Given real numbers a, b, c, with a+ b+ c= 0, prove that a 3 + b3 + c3 > 0 if and only if a5 + b5 + c5 > 0. (b) Given real numbers a, b, c, d, witha+ b+ c+ d= 0, prove that a 3 + b3 + c3 + d3 > 0 if and only if a5 + b5 + c5 + d5 > 0. 4. The real number xbetween 0 and 1 has decimal representation 0∙a 1a 2a 3a 4 . . . with the following property: the number of distinctblocks of the form a ka k+1 a k+2 . . . a k+2003 , as kranges through all positive integers, is less than or equal to 2004. Prove that xis rational.