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 hexagonal dart board
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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?
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Wrapping presents
Six squares are wrapped together with red ribbons. Do they cross at a single point?
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Tip the scale
Two equilateral triangles and a square. What’s the angle α?
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The cheese dome
A semicircle and two tangent circles (tangency point shown). What is the angle α?
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Hidden circle II
What is the locus of orange points for which the purple lines are parallel? The fixed cevians of the triangle BCD are angle bisectors.
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Half right
A right triangle with two angle bisectors. What is red : blue?
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Rolling level
Blue and green circles with tangents. The green tangent line goes through the blue tangent point and the points IJH are collinear. Show that the blue and orange lines are parallel.
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Your turn
A regular hexagon of which the centre is a vertex of an equilateral triangle. What is the angle α?