Maths Olympiad

Technology Mathematics Olympiad 2008-2009 (1) permit D,E,F be the points on the sides BC, CA, AB of the ?ABC where the in order of the triangle touches the sides. If AD = BE = CF designate that ?ABC is equilateral. (2) permit C1 and C2 be two cirles intersecting at the points A and B and let C0 be a circle through A with center at B. Determine (with proof) the conditions downstairs which the common chord of C0 and C1 is tangent to C2 . 1 1 (3) Let n ? 2 be an integer. stress that the sum 1 + 2 + 1 + . . . + n is not 3 an integer. (4) In a room there argon 10 race, none of whom are older than 60 (ages given in integral total only) but each of whom is atleast 1 years old. Prove that one can always ?nd two groups of batch (with no common person), the sum of whose ages is the same. Can 10 be replaced by a smaller number? Justify your answer. (5) Let n ? 2 be an integer such that 2n + n2 is a prime. Prove that n ? 3 (mod 6) (6) Suppose that n2 +1 people are lined up shoulder to shoulder in a straight line.



Show that it is always possible to spot n + 1 of them to take one step front so that seeing from left to right their heights are increasing or decreasing. (7)Three players A,B and C take turns in throwing a dice in order ABC,ABC,... What is the probability that (i) A is the sulphur player to get a six for the ?rst duration? (ii) A is the last player to get a six for the ?rst time? (8) Let a,b and c be positive real numbers such that abc = 1. Prove that a3 (b 1 1 3 1 + 3 + 3 ? + c) b (c + a) c (a + b) 2

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