<p>Let \(n \geq 2\) be an integer. Take \(n\) distinct points on a circle and join each pair of points by a line segment. Color 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 ________.</p>
Step-by-Step Solution
Key Concept: Total line segments connecting n points is C(n,2) = n(n-1)/2. Adjacent pairs on circle = n (blue). Red segments = n(n-1)/2 - n. Setting red = blue gives n(n-1)/2 - n = n, which simplifies to n² - 5n = 0.
<p><strong>Step 1:</strong> Count total line segments. For n distinct points, total line segments = C(n,2) = n(n-1)/2</p><p><strong>Step 2:</strong> Identify blue segments. Points on a circle form n pairs of adjacent points (consecutive on the circle), so blue segments = n</p><p><strong>Step 3:</strong> Count red segments. Red segments = Total - Blue = n(n-1)/2 - n = n(n-3)/2</p><p><strong>Step 4:</strong> Apply the condition that red = blue. Setting n(n-3)/2 = n, we get n-3 = 2 (dividing by n, valid since n ≥ 2), so n = 5</p><p><strong>Step 5:</strong> Verify: For n=5, Total = 5(4)/2 = 10; Blue = 5; Red = 10-5 = 5 ✓</p><p>∴ Answer: <strong>5</strong></p>
Correct Answer: 5