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Unformatted text preview: Wednesday, July 16, 2008 Problem 1. An acuteangled triangle ABC has orthocentre H . The circle passing through H with centre the midpoint of BC intersects the line BC at A 1 and A 2 . Similarly, the circle passing through H with centre the midpoint of CA intersects the line CA at B 1 and B 2 , and the circle passing through H with centre the midpoint of AB intersects the line AB at C 1 and C 2 . Show that A 1 , A 2 , B 1 , B 2 , C 1 , C 2 lie on a circle. Problem 2. (a) Prove that x 2 ( x 1) 2 + y 2 ( y 1) 2 + z 2 ( z 1) 2 ≥ 1 for all real numbers x , y , z , each different from 1, and satisfying xyz = 1 . (b) Prove that equality holds above for infinitely many triples of rational numbers x , y , z , each different from 1, and satisfying xyz = 1 . Problem 3. Prove that there exist infinitely many positive integers n such that n 2 + 1 has a prime divisor which is greater than 2 n + √ 2 n ....
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 Fall '11
 T.S.
 positive real numbers

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