$P(x)=(0.5-x)^{2015}+0.5$

IstekOlympiadTeam wrote: Find the solution of the functional equation $P(x)+P(1-x)=1$ with power $2015$ P.S: $P(y)=y^{2015}$ is also a function with power $2015$ I assume that you mean all polynomials of degree $2015$. In this case, let $P(x) = Q(1/2-x)+1/2$, then $Q$ is also a polynomial of degree $2015$ and we get $Q(1/2-x)=-Q(x-1/2)$ for all real $x$, which means that $Q$ is an odd function, i.e. all coefficients of even powers of $x$ vanish. On the other hand, if $Q$ is any such polynomial of degree $2015$, we easily see that $P(x) = Q(1/2-x)+1/2$ is a solution. So all solutions are the polynomials of this type.