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Problem 888

AIME late

Geometry Difficulty 5.7 Prove it

Problem 2. A convex figure $F$ has the property: any equilateral triangle with side 1 can be translated so that all its vertices lie on the contour of $F$ . Does it follow from this that $F$ is a circle?

This one wants a proof. Work it on paper, read the official solution, then mark yourself honestly — the ladder only means something if the record is true.

Official solution

Solution: We will prove that a figure with the given property is not necessarily a circle by considering a semicircle with radius 1.

It is sufficient to show that an equilateral triangle with side length 1 can be rotated by $120^{\circ}$ , with its vertices remaining on the boundary of the semicircle. This condition is clearly satisfied for a triangle with a vertex at the center

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of the semicircle, which rotates around this center.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.