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n-variable inequality with absolute values

Source: 2021 2nd OMpD L3 P3 - Brazil - Olimphída Matemáticos por Diversão

July 8, 2023
inequalitiesalgebran-variable inequalityabsolute value

Problem Statement

Let aa and bb be positive real numbers, with a<ba < b and let nn be a positive integer. Prove that for all real numbers x1,x2,…,xn∈[a,b]x_1, x_2, \ldots , x_n \in [a, b]: ∣x1−x2∣+∣x2−x3∣+⋯+∣xn−1−xn∣+∣xn−x1∣≤2(b−a)b+a(x1+x2+⋯+xn) |x_1 - x_2| + |x_2 - x_3| + \cdots + |x_{n-1} - x_n| + |x_n - x_1| \leq \frac{2(b - a)}{b + a}(x_1 + x_2 + \cdots + x_n) And determine for what values of nn and x1,x2,…,xnx_1, x_2, \ldots , x_n the equality holds.
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