Problem 20 · 2025 Math Kangaroo
Hard
Counting & Probability
careful-countingcomplementary-counting
Points B and C lie on the diameter of a semicircle with diameter AD, and points E, F, G and H lie on the arc. How many triangles exist whose vertices are three of these eight points?
Show answer
Answer: D — 52
Show hints
Hint 1 of 2
Count all triples of the 8 points, then subtract the degenerate ones.
Still stuck? Show hint 2 →
Hint 2 of 2
Four points (A, B, C, D) are collinear on the diameter, so triples chosen all from those four are not triangles.
Show solution
Approach: total triples minus collinear triples
- Choose any 3 of 8 points: C(8,3) = 56.
- The only collinear set is A, B, C, D on the diameter: C(4,3) = 4 bad triples.
- Triangles = 56 − 4 = 52.
Mark:
· log in to save