MathDB
Inequality Problem

Source: 2023 China TST Problem 21

April 1, 2023
inequalities

Problem Statement

Given integer n≥2n\geq 2. Find the minimum value of λ\lambda {}, satisfy that for any real numbers a1a_1, a2a_2, ⋯\cdots, an{a_n} and b{b}, λ∑i=1n∣ai−b∣+n∣∑i=1nai∣⩾∑i=1n∣ai∣.\lambda\sum\limits_{i=1}^n\sqrt{|a_i-b|}+\sqrt{n\left|\sum\limits_{i=1}^na_i\right|}\geqslant\sum\limits_{i=1}^n\sqrt{|a_i|}.
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