$ n$ sets $ S_1$, $ S_2$ $ \cdots$, $ S_n$ consists of non-negative numbers. $ x_i$ is the sum of all elements of $ S_i$, prove that there is a natural number $ k$, $ 1<k<n$, and: \[ \sum_{i=1}^n x_i < \frac{1}{k+1} \left[ k \cdot \frac{n(n+1)(2n+1)}{6} - (k+1)^2 \cdot \frac{n(n+1)}{2} \right]\] and there exists subscripts $ i$, $ j$, $ t$, and $ l$ (at least $ 3$ of them are distinct) such that $ x_i + x_j = x_t + x_l$.