Equal Number of Red and Blue Segments on Circle

To find the value of $n$, we analyze the total number of line segments that can be formed by joining $n$ distinct points on a circle.

1. Total Number of Line Segments: The total number of line segments formed by connecting any two of the $n$ distinct points is given by the combination formula: $\text{Total segments} = \binom{n}{2} = \frac{n(n-1)}{2}$

2. Number of Blue Segments: The line segments connecting adjacent points along the circle form an $n$-sided polygon. Therefore, the number of blue segments is equal to $n$: $\text{Blue segments} = n$

3. Number of Red Segments: The remaining line segments (which represent the diagonals of the $n$-gon) are colored red. Thus, the number of red segments is: $\text{Red segments} = \binom{n}{2} - n = \frac{n(n-1)}{2} - n$

4. Equating Red and Blue Segments: We are given that the number of red segments equals the number of blue segments: $\frac{n(n-1)}{2} - n = n$

   Adding $n$ to both sides gives: $\frac{n(n-1)}{2} = 2n$

   Since $n \ge 2$, we have $n \neq 0$. Dividing both sides of the equation by $n$: $\frac{n-1}{2} = 2$

   Multiplying both sides by $2$: $n - 1 = 4$ $n = 5$

Thus, the required value of $n$ is $5$.