1973 USAMO Problems/Problem 3

Problem

Three distinct vertices are chosen at random from the vertices of a given regular polygon of $(2n+1)$ sides. If all such choices are equally likely, what is the probability that the center of the given polygon lies in the interior of the triangle determined by the three chosen random points?

Solution

There are $\binom{2n+1}{3}$ ways how to pick the three vertices. We will now count the ways where the interior does NOT contain the center. These are obviously exactly the ways where all three picked vertices lie among some $n+1$ consecutive vertices of the polygon. We will count these as follows: We will go clockwise around the polygon. We can pick the first vertex arbitrarily ( $2n+1$ possibilities). Once we pick it, we have to pick $2$ out of the next $n$ vertices ( $\binom{n}{2}$ possibilities).

Then the probability that our triangle does NOT contain the center is $p = \frac{ (2n+1){\binom{n}{2}} }{ {\binom{2n+1}{3} } } = \frac{ (1/2)(2n+1)(n)(n-1) }{ (1/6)(2n+1)(2n)(2n-1) } = \frac{ 3(n)(n-1) }{ (2n)(2n-1) }$

And then the probability we seek is $1-p = \frac{ (2n)(2n-1) - 3(n)(n-1) }{ (2n)(2n-1) } = \frac{ n^2+n }{ 4n^2 - 2n } = \boxed{\frac{n+1}{4n-2}}$

Perplexing Edge Case

However, for $n = 1$ , (a rare edge case), the circumcenter of the triangle must lie in the center of the triangle, which is notably difficult to prove if not assumed, resulting in a probability of $1$ .

Observe that the probability tends to $\frac{1}{4}$ as $n$ approaches infinity. Why does this occur?

Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.

1973 USAMO (Problems • Resources)
Preceded by Problem 2	Followed by Problem 4
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1973 USAMO Problems/Problem 3

Contents

Problem

Solution

Perplexing Edge Case

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