2011 AIME I Problems/Problem 10

Problem

The probability that a set of three distinct vertices chosen at random from among the vertices of a regular n-gon determine an obtuse triangle is $\frac{93}{125}$ . Find the sum of all possible values of $n$ .

Solution

This is not complete and may not be correct. triangle is obtuse $\Longleftrightarrow$ there exists $\frac{n}{2}$ consecutive points that are not chosen. (i.e. all 3 points of the triangle are on the same half of the n-gon.

The probability of this happening is obviously lesser than $\frac{1}{2}$ , but $\frac{93}{125}>\frac{1}{2}$ . Thus there is no such possible n-gon?

2011 AIME I (Problems • Answer Key • Resources)
Preceded by Problem 9	Followed by Problem 11
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

Page

Toolbox

Search

2011 AIME I Problems/Problem 10

Problem

Solution

See also