Language: English

Day: 1

49th INTERNATIONAL MATHEMATICAL OLYMPIAD MADRID (SPAIN), JULY 10-22, 2008

Wednesday, July 16, 2008 Problem 1. An acute-angled triangle ABC has orthocentre H. The circle passing through H with centre the midpoint of BC intersects the line BC at A1 and A2 . Similarly, the circle passing through H with centre the midpoint of CA intersects the line CA at B1 and B2 , and the circle passing through H with centre the midpoint of AB intersects the line AB at C1 and C2 . Show that A1 , A2 , B1 , B2 , C1 , C2 lie on a circle. Problem 2. (a) Prove that y2 z2 x2 + + ≥1 (x − 1)2 (y − 1)2 (z − 1)2 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 n2 + 1 has a prime √ divisor which is greater than 2n + 2n.

Language: English

Time: 4 hours and 30 minutes Each problem is worth 7 points

Language: English

Day: 2

49th INTERNATIONAL MATHEMATICAL OLYMPIAD MADRID (SPAIN), JULY 10-22, 2008

Thursday, July 17, 2008 Problem 4. Find all functions f : (0, ∞) → (0, ∞) (so, f is a function from the positive real numbers to the positive real numbers) such that f (w)
2

+ f (x)

2

f (y 2 ) + f (z 2 )

=

w 2 + x2 y2 + z2

for all positive real numbers w, x, y, z, satisfying wx = yz. Problem 5. Let n and k be positive integers with k ≥ n and k − n an even number. Let 2n lamps labelled 1, 2, . . . , 2n be given, each of which can be either on or off. Initially all the lamps are off. We consider sequences of steps: at each step one of the lamps is switched (from on to off or from off to on). Let N be the number of such sequences consisting of k steps and resulting in the state where lamps 1 through n are all on, and lamps n + 1 through 2n are all off. Let M be the number of such sequences consisting of