Further correct solutions came from Jens Voss, Sudipta Das, Francisco de Leon-Sotelo y Esteban, Ivan Lisac, Edward Lee, Michael Lynch, Philippe Fondanaiche.

The probability is (4Ö2 - 5)/3 » 21.89%, or about 1 in 5.

Place a coordinate system on the square as in the sketch on the left.  Because of the symmetry of the problem, we can restrict ourselves to the shaded segment of the square.  Let P = (x,y) be a point in this segment.  The distance from P to the line along the top is 1 - y while the distance to the origin is Ö(x² + y² ).  Equating and solving gives the equation y = (1 - x² )/2 for the points equidistant from both. (Of course, you remembered from your geometry course that the equation would be a parabola, didn't you?) The points closer to the origin will lie inside this parabola.  As the parabola intersects the line y = x at the point Ö2 - 1, the area of the shaded region is

In the graph below, the points inside the region are closer to the origin than to the edge of the square.

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Page last updated 2 April 2001.

ã 2001 Alberto L Delgado