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How can you maximize your happiness under a budget? When does a function reach its minimum value? When does a curve change direction? The calculus of extrema explains these "extreme" situations.

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*

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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 \(a + b + c.\)

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

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

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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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