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rational function with f(1)+1>0

Source: Balkan Math Olympiad BMO 2003, problem 3

October 24, 2005
functionalgebra proposedalgebra

Problem Statement

Find all functions f:Q→Rf: \mathbb{Q}\to\mathbb{R} which fulfill the following conditions: a) f(1)+1>0f(1)+1>0; b) f(x+y)−xf(y)−yf(x)=f(x)f(y)−x−y+xyf(x+y) -xf(y) -yf(x) = f(x)f(y) -x-y +xy, for all x,y∈Qx,y\in\mathbb{Q}; c) f(x)=2f(x+1)+x+2f(x) = 2f(x+1) +x+2, for every x∈Qx\in\mathbb{Q}.
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