2021 Cayley Problem 24

In the diagram, PQ is a diameter of the circular base of the cylinder. RS is a diameter of the top face of the cylinder and is directly above PQ, as shown. Point U is on the circumference of the top face, halfway between R and S. Point T is on the cylinder and is directly above P. Suppose that QS = m and PT = n, where m and n are integers with 1 < n < m. If QU = 9\sqrt{33} and UT = 40, what is the remainder when the integer equal to QT^2 is divided by 100?

\textbf{(A)}\ 29\quad \textbf{(B)}\ 49\quad \textbf{(C)}\ 9\quad \textbf{(D)}\ 89\quad \textbf{(E)}\ 69

If there are no answer choices shown, enter a numerical answer.

Full credit to this problem is given to the CEMC, you may view all Cayley contests here.