2018 AMC 8 Problem 23
Attempt Problem 23 of the 2018 AMC 8 below, then check your answer against the video solution and professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2018 AMC 8 solutions, or check the answer key.
All problems are used with official legal permission of the Mathematical Association of America (MAA).
23.
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?
Answer: D
Small hint:
Use complementary counting: count triangles with no adjacent chosen vertices.
Big hint:
After fixing one vertex, count the positive gaps around the octagon between the chosen vertices.
Video solution:
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Written solution:
Count the complement: triangles with no two chosen vertices adjacent. Starting at a chosen vertex, let be the positive numbers of unchosen vertices in the three gaps. Then , which has positive solutions.
There are choices for the starting vertex, and each triangle is counted from each of its vertices, so the complement contains triangles. Out of total triangles, the desired probability is
Thus, D is the correct answer.
Problem 23 in other years
- 1985 AMC 8
- 1986 AMC 8
- 1987 AMC 8
- 1988 AMC 8
- 1989 AMC 8
- 1990 AMC 8
- 1991 AMC 8
- 1992 AMC 8
- 1993 AMC 8
- 1994 AMC 8
- 1995 AMC 8
- 1996 AMC 8
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- 1998 AMC 8
- 1999 AMC 8
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