IMO 08 ngày2 Q4:Find all functions $f: (0, \infty) \mapsto (0, \infty) $ (so f is a function from the positive real numbers) such that $\frac {\left( f(w) \right)^2 + \left( f(x) \right)^2}{f(y^2) + f(z^2) } = \frac {w^2 + x^2}{y^2 + z^2} $ for all positive real numbers w,x,y,z, satisfying wx = yz. Q5:Let n and k be positive integers with k≥ n and k-n an even number. Let 2n lamps labeled 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 number of such sequences consisting of k steps, resulting in the state where lamps 1 through n are all on, and lamps n+1 tgrough 2n are all off, but where none of the lamps n+1 through 2n is ever switched on. Determine N/M Q6: Let ABCD be a convex quadrilateral with BA different from BC. Denote the incircles of triangles ABC and ADC by k1 and k2 respectively. Suppose that there exists a circle k 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 k1 and k2 intersects on k. [RIGHT][I][B]Nguồn: MathScope.ORG[/B][/I][/RIGHT] __________________ Tiếu ngạo giang hồ Tóc bạc đi rồi ai có hay Giang hồ phiêu bạt bao ngày nay Rút kiếm khua sầu, sầu chén rượu Giương đao chém hận, hận cơn say Gió đông bước mãi giật mình lạnh Chốn cũ ngoảnh nhìn xa mưa bay Lá rơi bên mộ người tráng sĩ Lãng khách cười vang hát khúc này. |