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## Crazy hat

The diameter of the semicircle is a side of a rectangle. Show that for any such rectangle, and any point along its top edge, the configuration produces three collinear red dots.

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## Bottoming out

Two squares and a circle. What’s the angle α?

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## The tipi entrance

A semicircle with three tangents. Prove that purple : green = red : yellow.

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## Fan of cards

N coloured rectangular cards are arranged with a common vertex and common angle α between cards. If a single card has perimeter P, what is the perimeter of the coloured region, in terms of N, P, and α?

Extra condition: N and α are such that at least two edges of the bottom card are visible.

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## The telescope

The odd regular polygons are stacked together, in increasing order, zigzagging to form a telescope that extends forever! What is the limiting angle (with respect to the horizontal) at which the telescope points?

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## The weight lifter

Express the x coordinate of E in terms of x coordinates of A, B, F, and G. The two black lines are supposed to be parallel to the x-axis.

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## Encircling an anvil

Two equilateral triangles, one upside down, have boundaries meeting at 4 points, 3 of which are marked in red. Show that the circle through the red points also passes through the centre point of the blue triangle.

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## Baby block II

A square containing another square with an extended side. What is tan(α) in terms of the red and green segment lengths?

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