Mock AIME I 2012 Problems/Problem 1
A circle of maximal area is inscribed in the region bounded by the graph of and the -axis. The radius of this circle is , where , , and are integers and and are relatively prime. What is ?
Let be the circle of maximal area, and be the given parabola. By symmetry, the center of will be on the axis of , at . Because is tangent to the -axis, the y-coordinate of its center will be at (where is the radius). So has equation . Now suppose that is one of the two intersections of and . Then Adding these two equations and simplifying gives . By symmetry, there should only be one solution for , so the discriminant of this quadratic in is zero: . The answer is .