2021 IMO Problems/Problem 2

Problem

Show that the inequality $\sum_{i=1}^n \sum_{j=1}^n \sqrt{|x_i-x_j|} \le \sum_{i=1}^n \sum_{j=1}^n \sqrt{|x_i+x_j|}$ holds for all real numbers $x_1,x_2,\dots,x_n$ .

Solution

then, since $\sqrt{x}\geq 0, \sum_{i=1}^{n}\sum_{j=1}^{n}(\sqrt{x_i-x_j}^4)\leq \sum_{i=1}^{n}\sum_{j=1}^{n}(\sqrt{x_i+x_j}^4)$ then, $\sum \sum x_i^2+x_j^2-2x_ix_j \leq \sum \sum x_i^2+x_j^2+2x_ix_j \to \sum \sum 4x_ix_j\geq 0,$ therefore we have to prove that $\sum \sum a_ia_j\geq 0$ for every list [Xi], and we can describe this to $\sum \sum a_ia_j=\sum a_i^2 + \sum\sum a_ia_j(i\neq j)$ we know that $\frac{a_i^2}{2}+\frac{a_j^2}{2} \geq |a_ia_j|$ therefore, $a_i^2+a_j^2 \geq -(a_ia_j+a_ja_i)$ $\to \sum a_i^2 + \sum\sum a_ia_j \geq 0$ $Q.E.D.$ --Mathhyhyhy