Area Between Curves

Area Between Curves: Level 5 Challenges


The ends of a stiff bar AB\overline{AB} of length 4 slide freely inside a parabolic track (specifically, the parabola y=x2y=x^2). As they do, the midpoint MM of that bar traces a curve. Find the area of the region between the parabola and the curve traced by MM.

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 ln(a+b)+bc\frac{\ln(a+\sqrt{b})+\sqrt{b}}{c} where a,ba,b and cc are positive integers with bb being square-free. Find a+b+ca+b+c.

Let PP be a point (other than the origin) lying on the parabola y=x2y = x^{2}. The normal line to the parabola at PP will intersect the parabola at another point QQ. The minimum possible value for the area bounded by the line PQPQ and the parabola is ab\dfrac{a}{b}, where aa and bb are positive coprime integers. Find a+ba + 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 PP that is 14\dfrac{1}{4} units away from one of the edges of the strip.

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

A square 5×5 5 \times 5 is positioned in the 4th 4^\text{th} quadrant intersecting part of a circle of radius 55 and centered at (2,0) (-2, 0) . The square is rotated counter-clockwise about the origin.

Through what angle (in degrees) does the square have to be rotated until 12 \frac 1 2 of its area intersects with the circle?

[Answer to nearest 3 decimal places]


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