AT-OEMO-FED1 2011 Problem 2

For a positive integer $k$ and real numbers $x$ and $y$ , let $$f_k(x,y)=(x+y)-\left(x^{2k+1}+y^{2k+1}\right)\text{.}$$ If $x^2+y^2=1$ , then determine the maximal possible value $c_k$ of $f_k(x,y)$ .