Extrema: Level 3 Challenges


A circle rests in the interior of the parabola with equation \(y=x^2\) 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 \(S\) be the maximum possible area of a right triangle that can be drawn in a semi-circle of radius \(1\), where one of the legs (and not the hypotenuse) of the triangle must lie on the diameter of the semicircle.

If \(S = \dfrac{a\sqrt{b}}{c},\) where \(a,c\) are positive coprime integers and \(b\) is a positive square-free integer, find \(a + b + c.\)

How many real values of \(x\) satisfy the equation

\[\large {x}^{2}-{2}^{x}=0? \]

Suppose that two people, A and B, walk along the parabola \(y=x^2 \) in such a way that the line segment \( L\) between them is always perpendicular to the line tangent to the parabola at A's position \((a,a^2)\) with \( a > 0 \). If B's position is \((b,b^2)\), what value of \(b\) minimizes \(L\)?

What is the \(least\) \(perimeter\) of an isosceles triangle in which a circle of radius \(\sqrt{3}\) can be inscribed ?

Note : The picture shown is a rough one.


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