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n-variable modular inequality

Source: Bulgaria National Olympiad 2023 Problem 5

April 9, 2023
algebran-variable inequalityinequalities

Problem Statement

For every positive integer nn determine the least possible value of the expression ∣x1∣+∣x1−x2∣+∣x1+x2−x3∣+⋯+∣x1+x2+⋯+xn−1−xn∣|x_{1}|+|x_{1}-x_{2}|+|x_{1}+x_{2}-x_{3}|+\dots +|x_{1}+x_{2}+\dots +x_{n-1}-x_{n}| given that x1,x2,…,xnx_{1}, x_{2}, \dots , x_{n} are real numbers satisfying ∣x1∣+∣x2∣+⋯+∣xn∣=1|x_{1}|+|x_{2}|+\dots+|x_{n}| = 1.
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