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Advanced

The new normal

Two equilateral triangles are placed as shown. Their centres are connected as well as their intersection points. What’s the angle α?

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Intermediate

The radical

Two intersecting circles and a common tangent. The two tangency points are shown in black. Prove that the three red points are collinear.

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Intermediate

Getting stuck

A regular pentagon and a circle through two of its vertices and its centre. What’s the angle α?

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Beginner

The dirty thirties

A triangle containing a semicircle and a circular sector centred in the right vertex. What is blue : green?

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Intermediate

Scratching the surface

Three tangent semicircles and two coloured triangles. What’s the proportion yellow : green?

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Advanced

The Moroni minimum

An arbitrary point on the parabola y=x2. Shown are the tangent in that point and a line segment perpendicular to it. What is the minimal height h?

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Intermediate

To the core

A non-isosceles triangle with a red cevian and three line segments to a point on the cevian. Prove that this point is the incentre of the triangle.

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Beginner

Family ties

Two regular pentagons share a vertex. What is the proportion red : green?

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Intermediate

The seventh bell

A regular hexagon shares a vertex with an equilateral triangle of area 7. What is the area of the hexagon?

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Beginner

Longing for spring

Four semicircles and three line segments. What’s the proportion yellow : orange?