International Mathematical Olympiad 1999 Problem 6

AlgebraFunctionalFunctions

Determine all functions f:R⟶Rf: \mathbf{R} \longrightarrow \mathbf{R} such that

f(x−f(y))=f(f(y))+xf(y)+f(x)−1f(x - f(y)) = f(f(y)) + x f(y) + f(x) - 1

for all real numbers x,yx, y.