Solution
Acute Triangles in a Dodecagon
Show the problem again
How many triangles whose three vertices are vertices of a regular 12-gon have all three angles strictly less than 90°? Give an integer.
Worked solution
The answer is 40. The vertices lie on the circumcircle. The three vertices split the 12 arcs into gaps $a, b, c \ge 1$ with $a + b + c = 12$, and the angle opposite a gap of $g$ steps is $g \times 15°$ (inscribed angle theorem). The triangle is acute iff every gap is less than 6, i.e. the centre lies strictly inside the triangle. Count ordered gap triples with $1 \le a, b, c \le 5$ summing to 12: by stars and bars and inclusion-exclusion, $\binom{11}{2} - 3\binom{6}{2} + 3\binom{1}{2} = 55 - 45 + 0 = 10$. Each triangle corresponds to 3 ordered gap triples starting from each of its vertices, and there are 12 starting vertices, so the count is $12 \times 10 / 3 = 40$. (Of the 220 triangles, 60 are right and 120 obtuse.)
Source: Original problem written for AxiomIQ.