- #1

- 18,196

- 7,797

## Main Question or Discussion Point

Submitted and judged by: @QuantumQuest

Solution credit awarded to: @ddddd28

1) In order for a solution to count, a full derivation or proof must be given. Answers with no proof will be ignored.

2) It is fine to use nontrivial results without proof as long as you cite them and as long as it is "common knowledge to all mathematicians". Whether the latter is satisfied will be decided on a case-by-case basis.

3) If you have seen the problem before and remember the solution, you cannot participate in the solution to that problem.

4) You are allowed to use google, wolframalpha or any other resource. However, you are not allowed to search the question directly. So if the question was to solve an integral, you are allowed to obtain numerical answers from software, you are allowed to search for useful integration techniques, but you cannot type in the integral in wolframalpha to see its solution.

Note: This is a "B" level challenge. We ask that advanced members make use of spoiler tags and allow for less experience members an opportunity to solve. Thanks!

Three different vertices are chosen randomly from the set of vertices of a regular polygon having ##2n + 1## sides. Assume that all choices have equal probability. What is the probability that the center of polygon lies in the interior of the triangle that is formed from the three randomly chosen vertices?

Solution credit awarded to: @ddddd28

RULES:RULES:

1) In order for a solution to count, a full derivation or proof must be given. Answers with no proof will be ignored.

2) It is fine to use nontrivial results without proof as long as you cite them and as long as it is "common knowledge to all mathematicians". Whether the latter is satisfied will be decided on a case-by-case basis.

3) If you have seen the problem before and remember the solution, you cannot participate in the solution to that problem.

4) You are allowed to use google, wolframalpha or any other resource. However, you are not allowed to search the question directly. So if the question was to solve an integral, you are allowed to obtain numerical answers from software, you are allowed to search for useful integration techniques, but you cannot type in the integral in wolframalpha to see its solution.

**CHALLENGE:**Note: This is a "B" level challenge. We ask that advanced members make use of spoiler tags and allow for less experience members an opportunity to solve. Thanks!

Three different vertices are chosen randomly from the set of vertices of a regular polygon having ##2n + 1## sides. Assume that all choices have equal probability. What is the probability that the center of polygon lies in the interior of the triangle that is formed from the three randomly chosen vertices?

Last edited: