If $$[x]$$ represents the greatest integer less than or equal to $$x$$ then solve the equation $$x^{2011} - [x]^{2011} = (x - [x])^{2011}$$ in positive real numbers.

PLACEHOLDER ANSWER - the answer is a set of numbers. Let $$n = [x]$$ and $$f = x - [x]$$, so the equation is $$(n + f)^{2011} = n^{2011} + f^{2011}$$. If $$n \ge 1$$ and $$f > 0$$, the binomial expansion has extra positive terms, so the left side is larger. Hence $$n = 0$$ or $$f = 0$$, and the solutions are all $$x$$ with $$0 < x < 1$$ together with all positive integers.