## Task 6B - 171246B

a) Given a natural number $n \geq 2$. Let $u$ be the circumcircle of a regular $2 n$-gon $A_{0} A_{1} \ldots A_{2 n-1}$.

A set of three vertices $A_{i}, A_{j}, A_{k}$ of this $2 n$-gon is called one-sided if there is a semicircular arc $h$ on the circumference $u$, including its two endpoints, that contains $A_{i}, A_{j}$, and $A_{k}$.

Determine the probability $w$ that a randomly chosen set $M=\left\{A_{i}, A_{j}, A_{k}\right\}$ of three vertices is one-sided.

b) Determine the limit $\lim _{n \rightarrow \infty} w_{n}$ if it exists.

Hint:

If $m_{n}$ is the number of all sets that can be formed from three vertices $A_{i}, A_{j}, A_{k}$ of the $2 n$-gon, and $g_{n}$ is the number of one-sided sets among them, then the probability sought in a) is defined as $w_{n}=\frac{g_{n}}{m_{n}}$.
a) Let $B$ be the set of all one-sided triangles. Then $B$ is the disjoint union of the sets

$$
B_{i}=\left\{\left\{A_{i}, A_{j}, A_{k}\right\} \mid i<j<k \leq i+n\right\} \quad(i=0,1, \ldots, 2 n-1)
$$

where we set $A_{2 n}=A_{0}, A_{2 n+1}=A_{1}, \ldots, A_{2 n+n-1}=A_{n-1}$.

All sets $B_{i}$ have the same number of elements; it holds that $\left|B_{i}\right|=\binom{n}{2}$. Consequently,

$$
g_{n}=|B|=2 n\binom{n}{2}=n^{2}(n-1)
$$

From the vertices $A_{0}, A_{1}, \ldots A_{2 n-1}$, $m_{n}=\binom{2 n}{3}$ triangles can be formed. Thus, $w_{n}=\frac{n^{2}(n-1)}{\binom{2 n}{3}}=\frac{3 n}{2(2 n-1)}$ b) $\lim _{n \rightarrow \infty} w_{n}=\frac{3}{4}$