2025 AMC 10B Problem 6

The line y = \tfrac{1}{3}x + 1 divides the square region defined by 0 \le x \le 2 and 0 \le y \le 2 into an upper region and a lower region. The line x = a divides the lower region into two regions of equal area. Then a can be written as \sqrt{s} - t, where s and t are positive integers. What is s + t?

\textbf{(A)} 18 \qquad \textbf{(B)} 19 \qquad \textbf{(C)} 20 \qquad \textbf{(D)} 21 \qquad \textbf{(E)} 22

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.