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Inequality with absolute value

Source: Romania EGMO TST 2023 Day 2 P4

February 23, 2023
inequalitiesabsolute valueromaniaalgebra

Problem Statement

Let n⩾3n\geqslant 3 be an integer and a1,…,ana_1,\ldots,a_n be nonzero real numbers, with sum SS{}. Prove that ∑i=1n∣S−aiai∣⩾n−1n−2.\sum_{i=1}^n\left|\frac{S-a_i}{a_i}\right|\geqslant\frac{n-1}{n-2}.
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