The ends of a stiff bar of length 4 slide freely inside a parabolic track (specifically, the parabola ). As they do, the midpoint of that bar traces a curve. Find the area of the region between the parabola and the curve traced by .
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 where and are positive integers with being square-free. Find .
Let be a point (other than the origin) lying on the parabola . The normal line to the parabola at will intersect the parabola at another point . The minimum possible value for the area bounded by the line and the parabola is , where and are positive coprime integers. Find .
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 that is units away from one of the edges of the strip.
Let be the locus of all points such that if I draw the circle with center passing through , the entire circle can be drawn on the strip of paper. If the area of can be expressed by for relatively prime and square-free , then find .
A square is positioned in the quadrant intersecting part of a circle of radius and centered at . The square is rotated counter-clockwise about the origin.
Through what angle (in degrees) does the square have to be rotated until of its area intersects with the circle?
[Answer to nearest 3 decimal places]