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Problem 1058

AMC 12 late, AIME early

Algebra Difficulty 4.7 Prove it NMO Selection Tests for the Junior Balkan Mathematical Olympiad · Romania

On a circle are written several real numbers, of positive sum. Let $S$ be the largest and $s$ the least of the sums of consecutive numbers on the circle. Prove that $S + s > 0$ .

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

Denote by $T > 0$ the total sum of the numbers around the circle. Clearly $S \ge T > 0$ . If $s \ge 0$ , we are done. If $s < 0$ , the sum of the numbers which are not terms of $s$ is equal to $T - s$ . Since $S \ge T - s$ , then $S + s \ge T > 0$ , as needed.

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.