Let us denote F(n) = Σ_{i= 1 to n} i*(-1)ⁱ⁺¹

For example, F(4) = 1-2+3-4, F(5) = 1-2+3-4+5, F(6) = 1-2+3-4+5-6... and, so on.

(1) Determine the general form of two positive integers x and y that satisfy:
F(x) + F(y) + F(x+y) = 2012

(2) Can you explain why no positive integer solution exists whenever F(x) + F(y) + F(x+y) = 2011?