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3

Part of 2000 Federal Competition For Advanced Students, Part 2

Problems(2)

All functions so that f(x + f(y + z)) + f(f(x + y) + z) = 2y

Source: Austrian Mathematical Olympiad 2000, Part 2, D2, P3

6/28/2011
Find all functions f:R→Rf : \mathbb R \to \mathbb R such that for all reals x,y,zx, y, z it holds that f(x+f(y+z))+f(f(x+y)+z)=2y.f(x + f(y + z)) + f(f(x + y) + z) = 2y.
functionalgebra unsolvedalgebra
All real solutions of the absolue value problem

Source: Austrian Mathematical Olympiad 2000, Part 2, D1, P3

6/28/2011
Find all real solutions to the equation ∣∣∣∣∣∣∣x2−x−1∣−3∣−5∣−7∣−9∣−11∣−13∣=x2−2x−48.| | | | | | |x^2 -x - 1| - 3| - 5| - 7| - 9| - 11| - 13| = x^2 - 2x - 48.
algebra proposedalgebra