Problem 27 · 2013 Math Kangaroo
Stretch
Counting & Probability
careful-countingcasework
Any three vertices of a regular 13-sided shape are joined up to form a triangle. How many of these triangles contain the circumcentre of the 13-sided shape?
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Answer: C — 91
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Hint 1 of 2
A triangle inscribed in a circle contains the centre exactly when none of its three arcs is a half-circle or more.
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Hint 2 of 2
It is easier to count the triangles that miss the centre, then subtract from the total C(13,3).
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Approach: complementary counting of centre-missing triangles
- Total triangles: C(13,3) = 286.
- A triangle misses the centre when all three vertices lie within a half-circle; counting these and subtracting leaves the ones containing the centre.
- The number containing the circumcentre is 91.
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