15. A1 (CZE) Find all functions \( f \) from the reals to the reals such that 
\[ f(f(x)+y)=2x+f(f(y)-x) \]
for all real \( x, y \).
15. We observe first that \( f \) is surjective. Indeed, setting \( y = -f(x) \) gives \( f(f(-f(x)) - x) = f(0) - 2x \), where the right-hand expression can take any real value. In particular, there exists \( x_0 \) for which \( f(x_0) = 0 \). Now setting \( x = x_0 \) in the functional equation yields \( f(y) = 2x_0 + f(f(y) - x_0) \), so we obtain 
\[ f(z) = z - x_0 \quad \text{for } z = f(y) - x_0 \]
Since \( f \) is surjective, \( z \) takes all real values. Hence for all \( z \), \( f(z) = z + c \) for some constant \( c \), and this is indeed a solution.