Let $n \ge 2$ be an integer. Take $n$ distinct points on a circle and join each pair of points by a line segment. Colour the line segment joining every pair of adjacent points by blue and the rest by red. If the number of red and blue line segments are equal, then the value of $n$ is

Let n ≥ 2 be an integer. Take n distinct points on a circle and join each pair of points by a line segment. Colour the line segment joining every pair of adjacent points by blue and the rest by red. If the number of red and blue line segments are equal, then the value of n is $n\ge 2,\quad n \text{ distinct points on a circle}$ $\text{all } \binom{n}{2} \text{ pairs joined by segments}$ $\text{adjacent pairs } (n)\to\text{blue},\quad \text{the rest}\to\text{red}$ $\text{Given: } \#\text{red}=\#\text{blue},\qquad \text{find } n$ $\#\text{blue}=n\quad(\text{the } n \text{ adjacent sides})$ $\text{total segments}=\binom{n}{2}=\dfrac{n(n-1)}{2}$ $\text{blue}=n\quad(\text{the } n \text{ sides of the } n\text{-gon})$ $\text{red}=\binom{n}{2}-n=\dfrac{n(n-1)}{2}-n$ $\#\text{red}=\#\text{blue}:\quad \binom{n}{2}-n=n$ $\Rightarrow\ \dfrac{n(n-1)}{2}=2n$ $n(n-1)=4n$ $n\ne 0\ \Rightarrow\ n-1=4$ $n=5$ $n=5:\ \binom{5}{2}=10,\quad \text{blue}=5,\quad \text{red}=5\ \checkmark$ $\boxed{n=5}$