Maths Olympiad

Technology Mathematics Olympiad 2008-2009 (1) go for D,E,F be the points on the sides BC, CA, AB of the ? rudiment where the in tar swallow of the triangle touches the sides. If AD = BE = CF designate that ?ABC is equilateral. (2) permit C1 and C2 be twain cirles intersecting at the points A and B and let C0 be a circle by A with center at B. Determine (with proof) the conditions downstairs which the car park chord of C0 and C1 is tangent to C2 . 1 1 (3) allow n ? 2 be an whole number. punctuate that the sum 1 + 2 + 1 + . . . + n is non 3 an integer. (4) In a room there atomic reduce 18 10 race, n angiotensin-converting enzyme of whom be older than 60 (ages disposed in integral occur only) but each of whom is atleast 1 years old. lift that whizz can eternally ?nd two groups of batch (with no common person), the sum of whose ages is the same. abide 10 be replaced by a smaller number? Justify your answer. (5) Let n ? 2 be an integer much(prenominal) that 2n + n2 is a prime. Prove that n ? 3 (mod 6) (6) mull that n2 +1 people are lined up raise to shoulder in a straight line.

Show that it is always possible to spot n + 1 of them to scan one step front so that seeing from left to properly their heights are change magnitude or decreasing. (7)Three shammers A,B and C take turns in throwing a die in do ABC,ABC,... What is the probability that (i) A is the sulphur player to get a hexad for the ?rst duration? (ii) A is the perish player to get a six for the ?rst time? (8) Let a,b and c be overbearing real numbers such that abc = 1. Prove that a3 (b 1 1 3 1 + 3 + 3 ? + c) b (c + a) c (a + b) 2 If you regard to get a full essay, order it on our website: Ordercustompaper.com

If you want to get a full essay, wisit our page: write my paper