KURSCHAK 2013 Problem 3

Is it true that for integer $n\ge 2$ , and given any non-negative reals $\ell_{ij}$ , $1\le i<j\le n$ , we can find a sequence $0\le a_1,a_2,\ldots,a_n$ such that for all $1\le i<j\le n$ to have $|a_i-a_j|\ge \ell_{ij}$ , yet still $\sum_{i=1}^n a_i\le \sum_{1\le i<j\le n}\ell_{ij}$ ?