2018 AMC 8
Complete problem set with solutions and individual problem pages
From a regular octagon, a triangle is formed by connecting three randomly chosen vertices of the octagon. What is the probability that at least one of the sides of the triangle is also a side of the octagon?
- A.
- B.
- C.
- D.
- E.
Solution 1
Choose side "lengths" for the triangle, where "length" is how many vertices of the octagon are skipped between vertices of the triangle, starting from the shortest side, and going clockwise, and choosing if the triangle is isosceles: , where either [ and ] or [ (but this is impossible in an octagon)].
Options are: with in { 0,5 ; 1,4 ; 2,3 ; 3,2 ; 4,1 }, and with { 1,3 ; 2,2}. of these have a side with length 1, which corresponds to an edge of the octagon. So, our answer is
Solution 2
We will use constructive counting to solve this. There are cases: Either all points are adjacent, or exactly points are adjacent.
If all points are adjacent, then we have choices. If we have exactly adjacent points, then we will have places to put the adjacent points and places to put the remaining point, so we have choices. The total amount of choices is .
Thus, our answer is .
