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
Tag: 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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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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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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The Cheshire Cat grin
What is the total area of the green teeth relative to the large circle area? (Assume the inner circle radius is 80% of the outer. The 32 lines forming the tooth sides are concurrent, and spaced at equal angles apart.)
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The high-water mark
Two equilateral triangles on a blue line segment. Prove that the red line segment is parallel to it.
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Little red square
Two squares and a circle. The line through the big square centre and small square vertex is tangent to the circle. What is the red area?
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All-seeing eye
If the radius shrinks by the factor 1/2 for each smaller circle, what fraction of the whole area is represented by all the blue crescents (separated by the red crescents)? Assume the circles keep going inward forever.
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The ball pit
Two rectangles contain 6 congruent circles. The common rectangle side passes through the top centre. What’s yellow : red?