A correct solution was submitted by Adam Florzak.  Further correct solutions came from Bill Evans, Tomonori Ikeda, Surendra Dhir, Burkart Venzke, Mark Taurence, Philippe Fondanaiche, Sudipta Das, Francesc Suñol.

The equation of the circle is x² + y² = 1 while that of the parabola is y = Ax² + B, for some constants, A and B, to be determined.  At their points of intersection, the two graphs are tangent.  From the equation of the parabola we get y' = 2Ax, while that of the circle gives 2x + 2y y' = 0 or y' = -x/y; equating gives y = -1/2A. To find the points of intersection of the graphs we solve the equation A(1 - y²) + B = y for y.  This yields the relation B = -A - 1/(4A).

Next we find the area inside the parabolic region.  You can determine the points of intersection of the parabola and the line y = 1 easily from the equation of the parabola and the relation just determined. The point in the first quadrant lies at (1 + 1/(2A), 1), so the area to minimize is

ó1+1/(2A) õ0

1 - (Ax2 + B) dx = 0,

So A = 1 and B = -5/4.

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