2011 AMC 12A Problem 25

Triangle ABC has \angle BAC = 60^{\circ}, \angle CBA \leq 90^{\circ}, BC=1, and AC \geq AB. Let H, I, and O be the orthocenter, incenter, and circumcenter of \triangle ABC, respectively. Assume that the area of pentagon BCOIH is the maximum possible. What is \angle CBA?

\textbf{(A)}\ 60^{\circ} \qquad \textbf{(B)}\ 72^{\circ} \qquad \textbf{(C)}\ 75^{\circ} \qquad \textbf{(D)}\ 80^{\circ} \qquad \textbf{(E)}\ 90^{\circ}

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