A circle rests in the interior of the parabola with equation so that it is tangent to the parabola at two points. How much higher is the center of the circle than the points of tangency?
Source: Mandelbrot #2
Let be the maximum possible area of a right triangle that can be drawn in a semi-circle of radius , where one of the legs (and not the hypotenuse) of the triangle must lie on the diameter of the semicircle.
If where are positive coprime integers and is a positive square-free integer, find
How many real values of satisfy the equation
Suppose that two people, A and B, walk along the parabola in such a way that the line segment between them is always perpendicular to the line tangent to the parabola at A's position with . If B's position is , what value of minimizes ?
What is the of an isosceles triangle in which a circle of radius can be inscribed ?
Note : The picture shown is a rough one.