Problem 26 · 2021 Math Kangaroo
Stretch
Counting & Probability
careful-countingcasework
On a circle 15 points are equally spaced. We can form triangles by joining any 3 of these. Congruent triangles, by rotation or reflection, are counted as only one triangle. How many different triangles can be drawn?
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Answer: A — 19
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Hint 1 of 2
A triangle on the circle is described by the three gaps between its chosen points, which add to 15.
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Hint 2 of 2
Counting up to rotation and reflection means counting unordered gap-triples, i.e. partitions of 15 into three parts.
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Approach: count partitions of 15 into three positive parts
- An inscribed triangle is fixed (up to rotation/reflection) by the multiset of arc gaps a, b, c with a+b+c = 15 and each ≥ 1.
- So we count partitions of 15 into exactly three positive parts.
- There are 19 such partitions, hence 19 distinct triangles.
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