2019 AMC 12B Problem 8

Since $f(1-x)=(1-x)^2x^2=f(x),$ we have $f\!\left(\tfrac{k}{2019}\right)=f\!\left(\tfrac{2019-k}{2019}\right).$ In the sum, the term with index $k$ has sign $(-1)^{k+1},$ while the term with index $2019-k$ equals it in value but has sign $(-1)^{2019-k+1}=(-1)^{k},$ the opposite. Every term cancels with its partner, so the total is $0.$ Thus, A is the correct answer.