Question

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

Answer 1

Number of line segments $={}^n{C}_2$
Number of blue lines (formed by joining adjacent points) $=n$
So,
${}^n{C}_2-n=n\Rightarrow \frac{n(n-1)}{2}=2n\Rightarrow n^2=5n\Rightarrow n=0,5$
Since, $n\ge 2,\therefore n=5$