If f(x+y , x-y) = xy , then the arithmetic mean of f(x,y) and f(y,x) is ? 

replace x by x+y/2 and y with x-y/2 then f(x,y)=(x+y/2)*(x-y/2) and we can clearly see that if 
x and y are replaced then it is -f(y,x)
f(x,y)+f(y,x)=0 therefore average is 0