Problem

Source: 2024CTST P21

Tags: inequalities, 2024 CTST



Let integer $n\ge 3,$ $\tbinom n2$ nonnegative real numbers $a_{i,j}$ satisfy $ a_{i,j}+a_{j,k}\le a_{i,k}$ holds for all $1\le i <j<k\le n$. Proof $$\left\lfloor\frac{n^2}4\right\rfloor\sum_{1\le i<j\le n}a_{i,j}^4\ge \left(\sum_{1\le i<j\le n}a_{i,j}^2\right)^2.$$Proposed by Jingjun Han, Dongyi Wei