MathDB
Adding and Subtracting Functions

Source: 2019 AMC12 B #8

February 14, 2019
AMCAMC 12AMC 12 Bfunction

Problem Statement

Let f(x)=x2(1x)2f(x) = x^{2}(1-x)^{2}. What is the value of the sum \begin{align*} f\left(\frac{1}{2019}\right)-f\left(\frac{2}{2019}\right)+f\left(\frac{3}{2019}\right)-&f\left(\frac{4}{2019}\right)+\cdots\\ &\,+f\left(\frac{2017}{2019}\right) - f\left(\frac{2018}{2019}\right)? \end{align*}
<spanclass=latexbold>(A)</span>0<spanclass=latexbold>(B)</span>120194<spanclass=latexbold>(C)</span>2018220194<spanclass=latexbold>(D)</span>2020220194<spanclass=latexbold>(E)</span>1<span class='latex-bold'>(A) </span>0\qquad<span class='latex-bold'>(B) </span>\frac{1}{2019^{4}}\qquad<span class='latex-bold'>(C) </span>\frac{2018^{2}}{2019^{4}}\qquad<span class='latex-bold'>(D) </span>\frac{2020^{2}}{2019^{4}}\qquad<span class='latex-bold'>(E) </span>1