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Inequality with Strange Condition

Source: 2024CTST P21

March 28, 2024
inequalities2024 CTST

Problem Statement

Let integer n≥3,n\ge 3, (n2)\tbinom n2 nonnegative real numbers ai,ja_{i,j} satisfy ai,j+aj,k≤ai,k a_{i,j}+a_{j,k}\le a_{i,k} holds for all 1≤i<j<k≤n1\le i <j<k\le n. Proof ⌊n24⌋∑1≤i<j≤nai,j4≥(∑1≤i<j≤nai,j2)2.\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
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