Correct and complete solutions were received from Jeremy Light, Patrick Shea, Andrew Osmun, Matt Gillespie, Adam Schroeder, Daniel Marquette,l Scott Hoffmeyer, Mike Paluszkiewicz, Karen Whitehead, Adam Brown.  Correct and complete solutions were also received from Gary Smith, Juan Carlos Marivela, Steven Young, Alejandro Vellano, Mohamed Saad, Lou Cairoli, Julien Santini, Alan O'Donnell, Jens Voß, Paul Botham, Burkart Venzke, Al Zimmermann, Marko Marevic, Bill Webb, Steve King, Nancy Schwarzkopf, Ron Welch, Nick McGrath.

If the circle is too large, there will be three points of intersection, as in the figure on the left.  The equation of the parabola is y = x²; that of the circle of radius r, centered at (0,r) is x² + (y - r)² = r².  The points of intersection occur when  y(y + 1 - 2r) = 0, which occur when y = 0 = x, or when y = 2r - 1, x = ±Ö(2r - 1), the first corresponding to the intersection at the origin and the others to the two points of intersection off the origin.  In our case, we are looking for all three points to coincide, which occurs when 2r - 1 = 0, and r = 1/2. 

An alternate solution is to compute that the radius of curvature of the parabola at the origin is 1/2. 

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ã2003 Alberto L. Delgado