_{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?