Problem
ALG-B2-M04-P017 Two zero sums
#17
★★★★★ Level 5 of 5
Let \(a_1\le a_2\le\cdots\le a_n\), \(b_1\le b_2\le\cdots\le b_n\), and \(\sum_{i=1}^n a_i=\sum_{i=1}^n b_i=0\). Prove \[\sum_{i=1}^n a_i b_i\ge0.\]
Hint. Apply Chebyshev in its general form.
Since the sequences are similarly ordered, Chebyshev gives \[\frac1n\sum_{i=1}^n a_i b_i\ge\left(\frac1n\sum_{i=1}^n a_i\right)\left(\frac1n\sum_{i=1}^n b_i\right)=0.\] Hence \(\sum a_i b_i\ge0\).
This is an olympiad pattern: zero-sum conditions are often hidden in sign problems.