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Advanced

The maxbox

A square containing a red square of variable size sharing a vertex with a blue rectangle. What is the maximal blue fraction?

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Intermediate

The motorcycle

Two congruent orange circles and a smaller blue one. Three common tangents are shown, one of which passes through a circle centre. What is the proportion blue : orange?

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Beginner

The pit

A regular hexagon, a square with its diagonals, an equilateral triangle and a circle. Prove that the hexagon and the circle are concentric.

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Intermediate

The triple airbag

A triangle containing three tangent semicircles with diameters a, b and c. Three line segments pass through the red tangency points. What is the triangle perimeter?

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Advanced

The grazing shot

A parabola and its directrix in red. Two tangents intersect at a point on the directrix. What’s the angle α?

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Intermediate

The iron curtain

A parallelogram ABCD with a circle and a diameter CE. Prove that point E is the orthocentre of triangle ABD.

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Intermediate

Worlds colliding

Two intersecting circles and a circle passing through both their centres and their intersection points. What’s the angle α?

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Beginner

The common triangle

Two semicircles contain a triangle. The centre of the right semicircle is shown. Prove that the triangle is isosceles.

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Intermediate

At navel level

Two equilateral triangles share a vertex. The centre of the larger one is aligned with the midpoint of a side of the smaller one. What is green : red?

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Intermediate

Sharing is caring

Three equilateral triangles and three shared vertices. What’s the proportion blue : green?