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 w^2+x^2 ----------------- = ------- f(y^2)+f(z^2) y^2+z^2 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 k steps, resulting in the state where lamps 1 through n are all on, and lamps n+1 through 2n are all of, but where none of the lamps n+1 through 2n is ever switched on. Determine N/M Problem 6 Let ABCD be a convex quadrilateral with BA different from BC. Denote the incircles of triangles ABC and ADC by ω_1 and ω_2 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 ω_1 and ω_2 intersect on ω. 真抱歉...擺了大烏龍@@ 剛剛得到的題目來源是假的... --

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