Area Between Curves

Area Between Curves: Level 5 Challenges


The ends of a stiff bar \(\overline{AB}\) of length 4 slide freely inside a parabolic track (specifically, the parabola \(y=x^2\)). As they do, the midpoint \(M\) of that bar traces a curve. Find the area of the region between the parabola and the curve traced by \(M\).

Assume that the bar slides infinitely in both directions.

If the average distance from a point randomly selected in the unit square to its center equals \[\frac{\ln(a+\sqrt{b})+\sqrt{b}}{c}\] where \(a,b\) and \(c\) are positive integers with \(b\) being square-free. Find \(a+b+c\).

Let \(P\) be a point (other than the origin) lying on the parabola \(y = x^{2}\). The normal line to the parabola at \(P\) will intersect the parabola at another point \(Q\). The minimum possible value for the area bounded by the line \(PQ\) and the parabola is \(\dfrac{a}{b}\), where \(a\) and \(b\) are positive coprime integers. Find \(a + b\).

Clarification: The normal line is the line perpendicular to the tangent line at a given point on a curve and which passes through the given point.

I have a infinitely long strip of paper that is one unit wide. On the paper is a point \(P\) that is \(\dfrac{1}{4}\) units away from one of the edges of the strip.

Let \(L\) be the locus of all points \(O\) such that if I draw the circle with center \(O\) passing through \(P\), the entire circle can be drawn on the strip of paper. If the area of \(L\) can be expressed by \(\dfrac{a\sqrt{b}}{c}\) for relatively prime \(a,c\) and square-free \(b\), then find \(a+b+c\).

A square \( 5 \times 5 \) is positioned in the \( 4^\text{th} \) quadrant intersecting part of a circle of radius \(5\) and centered at \( (-2, 0) \). The square is rotated counter-clockwise about the origin.

Through what angle (in degrees) does the square have to be rotated until \( \frac 1 2 \) of its area intersects with the circle?

[Answer to nearest 3 decimal places]


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