Problem 29 · 2024 Math Kangaroo
Stretch
Geometry & Measurement
careful-countingsymmetry
Twenty points are spaced equally around a circle. How many of the segments joining two of these points are longer than the radius of the circle but shorter than its diameter?
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Answer: C — 120
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Hint 1 of 2
A chord between points k steps apart has length 2R*sin(k*pi/20).
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Hint 2 of 2
Find which step counts k make this longer than R but shorter than the diameter 2R.
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Approach: convert the length condition to a range of step counts
- The chord length 2R*sin(k*pi/20) exceeds R when k >= 4, and is below the diameter for every k except the 10-step diameter.
- So k can be 4,5,6,7,8,9 (six values), each giving 20 chords by symmetry.
- That is 6 x 20 = 120 segments.
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