1974 USAMO Problems/Problem 3

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Problem

Two boundary points of a ball of radius 1 are joined by a curve contained in the ball and having length less than 2. Prove that the curve is contained entirely within some hemisphere of the given ball.

Solution

Draw a Great Circle containing the two points and the curve. Then connect the two points with a chord in the circle. The circle has radius 1, so the circle has diameter 2, so the two points are all but diametrically opposite. Therefore we can draw a diameter parallel to the chord but not on it.

Now take the plane through EF perpendicular to AG. This creates two hemispheres, and the curve is in one and only one of them.

See also