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The Bermuda Triangle

A triangle with an inscribed rectangle. What is the maximum red fraction?

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The rolling stones

If you roll a circle down a parabola, small circles can roll all the way down and up, whereas large circles get stuck between the sides at some point. What is the transition radius? In other words, what’s the maximum circle that keeps on rolling?

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The PowerPoint

A point lies 8 from the centre of a circle of radius 6. The angle α is variable. What’s the maximum triangle area?

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Don Quixote and the windmill

Don Quixote is approaching a windmill. The sails of length √2 are attached to a point that is 2 above eye level. What is the maximum vertical angle that Don Quixote perceives as he moves forward?

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Intermediate

Beware of the dots

Two congruent circles and a common tangent. A triangle connecting tangency points and the upper intersection point. What angle α maximises its area?

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Intermediate

The bowtie

A square and two congruent isosceles triangles with variable top angle. What is the maximum fraction shaded?

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The shark fin

An isosceles triangle is inscribed in a semicircle with one side along the diagonal and the top vertex somewhere on the semicircle. What’s the maximum fraction shaded?