数学试题讲解

设整数

n \geq 3

. 设

\frac{1}{2} n (n - 1)

个非负实数

a_{i, j} (1 \leq i < j \leq n)

满足对任意

1 \leq i < j < k \leq n

, 均有

a_{i, j} + a_{j, k} \leq a_{i, k}

. 求证:

[\frac{n^{2}}{4}] \sum_{1 \leq i < j \leq n} a_{i, j}^{4} \geq {(\sum_{1 \leq i < j \leq n} a_{i, j}^{2})}^{2} .