Bound on k extreme values
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Let $A = [a_1, a_2, \ldots, a_n]$ be $n$ numbers where $i < j \implies a_i \le a_j$.
Let $s = \sum_{i=1}^n a_i$, $x = \sum_{i=1}^k a_i$, $y = \sum_{i=n-k+1}^n a_i$. Then $x \le \frac{sk}{n}$ and $y \ge \frac{sk}{n}$.
Proof
\begin{align} & a_k \ge \frac{x}{k} \\ &\implies s = x + \sum_{i=k+1}^n a_i \ge x + (n-k)\frac{x}{k} = \frac{xn}{k} \\ &\implies x \le \frac{sk}{n} \end{align}
The proof of $y \ge \frac{sk}{n}$ is similar.
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