2018 AIME I Problems/Problem 7
A right hexagonal prism has height . The bases are regular hexagons with side length . Any of the vertices determine a triangle. Find the number of these triangles that are isosceles (including equilateral triangles).
We can consider two cases: when the three vertices are on one base, and when the vertices are on two bases.
Case 1: vertices are on one base. Then we can call one of the vertices for distinction. Either the triangle can have sides with 6 cases or with 2 cases. This can be repeated on the other base for cases.
Case 2: The vertices span two bases. WLOG call the only vertex on one of the bases . Call the closest vertex on the other base , and label clockwise . We will multiply the following scenarios by , because the top vertex can have positions and the top vertex can be on the other base. We can have , but we are not done! Don't forget that the problem statement implies that the longest diagonal in a base is and the height is , so is also correct! Those are the only three cases, so there are cases for this case.
In total there's cases.
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