[PDF] UKMT - BMO Round 2 - British Mathematical Olympiad 1993.pdf | Plain Text

BRITISH MATHEMATICAL OLYMPIAD Round 2 : Thursday, 11 February 1993 Time allowed Three and a half hours. Each question is worth 10 marks. Instructions •Ful l written solutions are required. 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 trying al l four problems. • The use of rulers and compasses is al lowed, but calculators 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. Before March, twenty students will be invited to attend the training session to be held at Trinity College, Cambridge (on 15-18 April). On the final morning of the training session, students sit a paper with just 3 Olympiad-style problems. The UK Team for this summer’s International Mathematical Olympiad (to be held in Istanbul, Turkey, July 13–24) 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 before leaving for Istanbul. Do not turn over until told to do so. B RITISH MATHEMATICAL OLYMPIAD 1. We usually measure angles in degrees, but we can use any other unit we choose. For example, if we use 30 ◦ as a new unit, then the angles of a 30 ◦ , 60 ◦ , 90 ◦ triangle would be equal to 1, 2, 3 new units respectively. The diagram shows a triangle AB Cwith a second triangle DE Finscribed in it. All the angles in the diagram are whole number multiples of some new (unknown unit); their sizes a, b, c, d, e, f , g, h, i, j, k, ℓ with respect to this new angle unit are all distinct. Find the smallest possible value of a+ b+ cfor which such an angle unit can be chosen, and mark the corresponding values of the angles a to ℓin the diagram. 2. Let m= (4 p − 1)/3, where pis a prime number exceeding 3. Prove that 2 m −1 has remainder 1 when divided by m. 3. Let Pbe an internal point of triangle AB Cand let α, β , γbe defined by α = 6 B P C −6 B AC, β =6 C P A −6 C B A, γ =6 AP B −6 AC B . Prove that P Asin 6 B AC sin α = P B sin 6 C B A sin β = P C sin 6 AC B sin γ . 4. The set Z(m, n ) consists of all integers Nwith mndigits which have precisely nones, ntwos, nthrees, . . .,n m s. For each integer N∈Z(m, n ), define d(N ) to be the sum of the absolute values of the differences of all pairs of consecutive digits. For example, 122313 ∈Z(3 ,2) with d(122313) = 1 + 0 + 1 + 2 + 2 = 6. Find the average value of d(N ) as Nranges over all possible elements of Z(m, n ).