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The flat earth

An observer is standing on a plane at h metres below the centre of a reflecting sphere of radius r. Another object on the plane is perceived at an angle α from the vertical. What is α of the horizon in terms of h and r?

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Intermediate

Z

A green quadrilateral of which two sides are tangent to a quarter circle of radius r. What is its area?

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Intermediate

Overlapping interests

A square, a rectangle and a circle that partially overlap. One square vertex coincides with a rectangle-circle tangency point. Prove that the area of the square equals that of the rectangle.

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Beginner

The common point

Two equilateral triangles share a vertex. What’s the angle α?

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The sliding square

A square of variable size inscribed in a semicircle of radius r. What is the maximum distance d of the lowest square vertex to the nearest semicircle corner?

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Two moons

A rectangle with two congruent circles, a semicircle and a common tangent. What is the proportion of side lengths a : b?

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Intermediate

The rising sun

A rectangle with a circle and a semicircle. What is the proportion red : blue in terms of a and b?

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Intermediate

The vanishing point

A square and an equilateral triangle. An extended side and diagonal meet in a point. What is the angle α?

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Beginner

The flying saucer

A quarter circle with four chords. What is the proportion purple : gold?

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Intermediate

The dance

A regular hexagon, two squares and a semicircle connecting two vertices. Is the red vertex located on the semicircle?