Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 4.8 AIME Prove it Saudi Arabia

Let $n$ be a positive integer and let $a_{1}, a_{2}, \ldots, a_{n}$ be any real numbers. Prove that there exists $m, k \in \{1,2, \ldots, n\}$ such that
$\left|\sum_{i=1}^{m} a_{i}-\sum_{i=m+1}^{n} a_{i}\right| \leq\left|a_{k}\right| .$

Solution

Denote
$S_{x}=\left|\sum_{i=1}^{x} a_{i}-\sum_{i=x+1}^{n} a_{i}\right| .$
Obviously $S_{0}=-S_{n}$ and $S_{x+1}-S_{x}=2 a_{x+1}$ . We may assume that $S_{n} \geq 0$ and consider $m$ for which $S_{m} \leq 0$ and $S_{m+1} \geq 0$ . Then
$\left|S_{m}\right|+\left|S_{m+1}\right|=\left|S_{m}-S_{m+1}\right|=2\left|a_{m+1}\right|$
so either $\left|S_{m}\right|$ or $\left|S_{m+1}\right|$ is at most $a_{m+1}$ which is at $\operatorname{most}_{\max _{i}}\left|a_{i}\right|=a_{k}$ . This completes the proof.

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