2008 IMO Problems
Problems of the 49th IMO 2008 Spain.
Let be the orthocenter of an acute-angled triangle . The circle centered at the midpoint of and passing through intersects line at points and . Similarly, define the points , , and .
Prove that six points , , , , and are concyclic.
Let be three real numbers, such that
(i) Prove that; .
(ii) Prove that for infinitely many triples of rational numbers , and .
Prove that there are infinitely many positive integers such that has a prime divisor greater than .
Find all functions (so is a function from the positive real numbers) such that
for all positive real numbes satisfying
Let and be positive integers with and an even number. Let lamps labelled 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 be the number of such sequences consisting of steps and resulting in the state where lamps through are all on, and lamps through are all off.
Let be number of such sequences consisting of steps, resulting in the state where lamps through are all on, and lamps through are all off, but where none of the lamps through is ever switched on.
Let be a convex quadrilateral with different from . Denote the incircles of triangles and by and respectively. Suppose that there exists a circle tangent to ray beyond and to the ray beyond , which is also tangent to the lines and .
Prove that the common external tangents to and intersect on .
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