International Mathematical Olympiad - IMO 2008

Language: English Day: 149th INTERNATIONAL MATHEMATICAL OLYMPIADMADRID (SPAIN), JULY 10-22, 2008Wednesday, July 16, 2008Problem 1. An acute-angled triangle ABC has orthocentre H. The circle passing through H withcentre the midpoint of BC intersects the line BC at A 1 and A 2 . Similarly, the circle passing throughH with centre the midpoint of CA intersects the line CA at B 1 and B 2 , and the circle passing throughH 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 thatx 2(x − 1) 2 + y 2(y − 1) 2 + z 2(z − 1) 2 ≥ 1for 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, eachdifferent from 1, and satisfying xyz = 1.Problem 3. Prove that there exist infinitely many positive integers n such that n 2 + 1 has a primedivisor which is greater than 2n + √ 2n.Language: EnglishTime: 4 hours and 30 minutesEach problem is worth 7 points