2018 AIME I Problems/Problem 7
Problem
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).
Solution
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.
See also
| 2018 AIME I (Problems • Answer Key • Resources) | ||
| Preceded by Problem 6 |
Followed by Problem 8 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
