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

Question: Why is the curve necessarily on a great circle? The curve is arbitrary–in space. Also, there is only one great circle through the two points, as three points determine a plane–a counterexample to this claim would then be readily found. 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 $EF$ parallel to the chord but not on it.

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

Solution 2

Let the curve intersect the sphere at points A and B. Let B' be the point diametrically opposite A, and let $Q$ be the plane through the center O of the sphere perpendicular to $BB'$ and passing through the midpoint of $BB'$. We claim that the curve must be in the hemisphere divided by $Q$ containing points A and B.

Assume the contrary and suppose that the curve intersects $Q$ at point C. Reflect the portion of the curve from C to B to obtain a same-length curve from C to B'. Then the length of the curve equals the length of the new curve from A to B', which is at least the distance from A to B', or 2. This is a contradiction to the claim that the length of the curve is less than 2, so the curve must be contained in the hemisphere divided by $Q$ containing points A and B.

Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.

See Also

1974 USAMO (ProblemsResources)
Preceded by
Problem 2
Followed by
Problem 4
1 2 3 4 5
All USAMO Problems and Solutions

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