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Advanced

Alien eye

Two quarter-circles fit inside a square, and five circles fit inside the overlapped region, centred. What is the ratio of the (straight line) segment lengths red to blue? (The segments connect to points of tangency.)

Categories
Intermediate

Pop you bubble

A square with a semicircle and a circle of equal radius. Their tangency point is shown. Prove that the red triangle is equilateral.

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Advanced

The ceiling lamp

A regular octagon with two diagonals and a square. Prove that the four red points are concyclic.

Categories
Beginner

Stay centred

Two squares share a vertex. Prove that the red point is the circumcentre of the red triangle.

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Advanced

Peeping out II

A unit square and a rectangle touching three square sides and passing through the midpoint of the upper side. What is the minimal blue area?

Categories
Beginner

Peeping out

A unit square and a rectangle. What is the blue area?

Categories
Beginner

Pythagoras times three

Three squares on a grid. What is the ratio of the areas red : green : purple?

Categories
Intermediate

Bottoming out II

A semicircle with three squares. What’s the angle α?

Categories
Beginner

The heptagon square

What is the angle α between the blue line and the baseline of the orange square?

Categories
Beginner

A golden year

Two squares, a rectangle and a semicircle. What is yellow : green?