Five squares in a configuration. What’s the angle α?
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Movie film II
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Downhill skiing
Choose any point D on the side AB of an isosceles triangle ABC, then extend the side AC beyond C to a point F with |CF| = |DB|. Show that the segment DF is cut exactly in two by the third side BC—that the orange and red trails are equal.
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Ellipsoidal laser
The inside surface of an ellipse is a perfect mirror. There is a pin-hole at an end of the diameter. Show that a light ray emitted from either focus will exit the enclosure via the pin-hole, perhaps after bouncing through the foci several times.
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Roll over Beethoven
Two regular hexagons share a vertex. If the red hexagon has area 4, what is the area of the blue one?
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Wrapping presents II
Six squares are placed around a triangle. What is the angle α?
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Corner pursuit
Four corners of a square pursue each other clockwise, and after each unit step mark out the new smaller square. Show that the squares sizes in the sequence decrease in size to 1 in the limit.
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Off the grid
A square on a grid containing a red triangle with an extended side. What is the red fraction?
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The hexagonal dart board
25 darts land inside a regular-hexagonal dart board with edge length the square root of 3. Show that at least five of the darts land inside the same unit circle.
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The mirror dome
A ray of light is emitted from a point on the wall of a mirror-covered semicircle and reflects four times as shown to return to the same spot. If the radius is 1, what is the length of the path?
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Folded napkin II
A unit square paper is folded as shown. What is the red area?