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Beginner

The voting pencil

A square and a rectangle share a vertex. Two triangle areas are given. What is the total area?

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Advanced

Out of focus

A semicircle with an inscribed triangle. Both circles are tangent to the semicircle, the diameter and one other triangle side. Prove that the line segment connecting the two shown tangency points is parallel to the diameter.

Categories
Beginner

The pentagon attack

Two regular pentagons and two diagonals. What’s the angle α?

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Intermediate

The red triangle

A square containing an isosceles triangle with given side lengths. What fraction is red?

Categories
Intermediate

As within, so without

A semicircle and a circle are tangent. The tangency point is shown. Prove that the two coloured triangles are similar.

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Beginner

Confined enclosure

A square containing four smaller squares. What fraction of its area is orange?

Categories
Beginner

Lean on me

Two squares and two diagonals. Prove that the angles α and β are equal.

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Advanced

The sacrosanct circles

Two tangent circles and two tangent line segments meeting in a point on the outer circle. The tangency points are connected by a line segment of length x. What’s x in terms of a and b?

Categories
Intermediate

The sliding ladder

A ladder seen from the side is sliding down a wall. Prove that a fixed point on the ladder traces part of an ellipse.

Categories
Intermediate

Check and balance

A square divided by three line segments. The red and blue areas are equal. What’s the length proportion a : b?