DK-GEORGMOHR 2024 Problem 5

Let $0 < x_1 < x_2 < ... < x_{2024} = 1$ be real numbers. A colouring of these numbers consists of painting $1012$ of the numbers red and the remaining $1012$ numbers blue. The red numbers are denoted in ascending order $r_1 < ... < r_{1012}$ , and the blue numbers in ascending order $b_1 <...< b_{1012}$ . Two examples of colourings are: $$\underbrace{x_1, . . . , x_{1012}}_{r_1,...,r_{1012}},\underbrace{x_{1013}, ... , x_{2024}}_{b_1,...,b_{1012}} \,\, and \underbrace{x_1}_{r_1}, \underbrace{x_2}_{b_1},\underbrace{x_3}_{r_2},\underbrace{x_4}_{b_2},...,\underbrace{x_{2023}}_{r_{1012}} ,\underbrace{x_{2024}}_{r_{1013}}.$$ For any colouring one considers the quantity $$S = r_1b_1 + r_2(b_2-b_1) + r_3(b_3 - b_2) +...+ r_{1012}(b_{1012} - b_{1011}).$$ a) Prove that one may choose the numbers $x_1, .., x_{2024}$ such that any colouring satisfies $S < \frac12$ . b) Prove that no matter how the numbers $x_1, .., x_{2024}$ are chosen, there exists a colouring so that $S > \frac{x_{2023}}{2}$ .