[PDF] UKMT - BMO Round 2 - British Mathematical Olympiad 2005.pdf | Plain Text

Supported by British Mathematical OlympiadRound 2 : Tuesday, 1 February 2005 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 (7-11 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 further training. The UK Team of 6 for this summer’s International Mathematical Olympiad (to be held in Merida, Mexico, 8 - 19 July) will then be chosen. Do not turn over until told to do so. Supported by 2005 British Mathematical Olympiad Round 2 1. The integer Nis positive. There are exactly 2005 ordered pairs ( x, y) of positive integers satisfying 1 x+ 1 y= 1 N. Prove that Nis a perfect square. 2. In triangle AB C,6 B AC = 120◦ . Let the angle bisectors of angles A, B andCmeet the opposite sides in D, EandFrespectively. Prove that the circle on diameter E Fpasses through D. 3. Let a, b, cbe positive real numbers. Prove that ³a b + b c+ c a´ 2 ≥ (a + b+ c)³ 1 a + 1 b+ 1 c´ . 4. Let X={A 1, A 2, . . . , A n} be a set of distinct 3-element subsets of { 1,2 , . . . , 36}such that i) A iand A j have non-empty intersection for every i, j. ii) The intersection of all the elements of Xis the empty set. Show that n≤ 100. How many such sets Xare there when n= 100?