International Math Olympiad Questions
IMO Problems 2008
IMO 2008 Problem 1
Let H be the orthocenter of an acute angled triangle ABC. The circle Γ centered at the midpoint of BC at points A₁, A₂. Similarly define the points B₁, B₂, C₁, and C₂.
Prove that the six points A₁, A₂, B₁, B₂, C₁, and C₂ are concyclic.
IMO 2008 Problem 2
(A) Prove that
x²/(x-1)² + y²/(y-1)² + z²/(z-1)² ≥ 1
for real numbers x, y, and z each different from 1 and satisfying the condition xyz = 1.
(B) Prove that equality holds above for infinitely many triplets of rational numbers x, y, and z each different from 1 and satisfying xyz = 1.
IMO 2008 Problem 3
Prove that there are infinitely many positive integers n such that n² + 1 has a prime divisor greater than 2n + (2n)1/2.
IMO 2008 Problem 4
Find all functions f: (0, ∞) → (0, ∞), such that
(f²(w) + f²(x))/(f²(y) + f²(z)) = (w² + x²)/(y² + z²), for all positive real numbers x, y, z, w, satisfying wx = yz.
IMO 2008 Problem 5
Let n and k be positive integers with k ≥ n and k-n an even number. Let 2n lamps labelled 1, 2, 3, …, 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 and from off to on).
(A) 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 to 2n are all off.
(B) Let M 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 to 2n are all off, but where none of the lamps n+1 through to 2n is ever switched on.
Determine the value of N/M.
IMO 2008 Problem 6
Let ABCD be a convex quadrilateral with BA ≠ BC. Denote the incircle of triangle ABC and ADC by ω₁, ω₂ respectively. Suppose that there exists a circle ω tangent to ray BA beyond A and to the ray BC beyond C, which is also tangent to the lines AD and CD.
Prove that the common external tangents to ω₁ and ω₂ intercept on ω.