Extrema: Level 3 Challenges


A circle rests in the interior of the parabola with equation y=x2y=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 SS be the maximum possible area of a right triangle that can be drawn in a semi-circle of radius 11, where one of the legs (and not the hypotenuse) of the triangle must lie on the diameter of the semicircle.

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

How many real values of xx satisfy the equation

x22x=0?\large {x}^{2}-{2}^{x}=0?

Suppose that two people, A and B, walk along the parabola y=x2y=x^2 in such a way that the line segment L L between them is always perpendicular to the line tangent to the parabola at A's position (a,a2)(a,a^2) with a>0 a > 0 . If B's position is (b,b2)(b,b^2), what value of bb minimizes LL?

What is the leastleast perimeterperimeter of an isosceles triangle in which a circle of radius 3\sqrt{3} can be inscribed ?

Note : The picture shown is a rough one.


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