2001 Cayley Problem 19

Three different numbers are chosen such that when each of the numbers is added to the average of the remaining two, the numbers 65, 69 and 76 result. The average of the three original numbers is

\textbf{(A)}\ 34\quad \textbf{(B)}\ 35\quad \textbf{(C)}\ 36\quad \textbf{(D)}\ 37\quad \textbf{(E)}\ 38

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.