[PDF] UKMT - BMO Round 2 - British Mathematical Olympiad 1996.pdf | Plain Text

BRITISH MATHEMATICAL OLYMPIAD Round 2 : Thursday, 15 February 1996 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 (28–31 March). On the final morning of the training session, students sit a paper with just 3 Olympiad-style problems. The UK Team - six members plus one reserve - for this summer’s International Mathematical Olympiad (to be held in New Delhi, India, 7–17 July) will be chosen immediately thereafter. Those selected will be expected to participate in further correspondence work between April and July, and to attend a short residential session 30 June–4 July before leaving for India. Do not turn over until told to do so. B RITISH MATHEMATICAL OLYMPIAD 1. Determine all sets of non-negative integers x, yand zwhich satisfy the equation 2x + 3 y = z2 . 2. The sides a, b, candu, v, w of two triangles AB CandU V W are related by the equations u(v + w− u) = a2 , v (w + u− v) = b2 , w (u + v− w) = c2 . Prove that triangle AB Cis acute-angled and express the angles U, V , W in terms of A, B , C. 3. Two circles S 1 and S 2 touch each other externally at K; they also touch a circle Sinternally at A 1and A 2respectively. Let P be one point of intersection of Swith the common tangent to S 1 and S 2 at K. The line P A 1meets S 1 again at B 1, and P A 2meets S 2 again at B 2. Prove that B 1B 2 is a common tangent to S 1 and S 2. 4. Let a, b, canddbe positive real numbers such that a+ b+ c+ d= 12 and abcd= 27 + ab+ac +ad +bc +bd +cd. Find all possible values of a, b, c, dsatisfying these equations.