2024 AMC 10A Problem 17

Two teams are in a best-two-out-of-three playoff: the teams will play at most 3 games, and the winner of the playoff is the first team to win 2 games. The first game is played on Team A's home field, and the remaining games are played on Team B's home field. Team A has a \frac{2}{3} chance of winning at home, and its probability of winning when playing away from home is p. Outcomes of the games are independent. The probability that Team A wins the playoff is \frac{1}{2}. Then p can be written in the form \frac{1}{2}(m - \sqrt{n}), where m and n are positive integers. What is m+n?

\textbf{(A)}~10\qquad\textbf{(B)}~11\qquad\textbf{(C)}~12\qquad\textbf{(D)}~13\qquad\textbf{(E)}~14

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