Problem
ALG-B2-M01-P013 Two positive arcs
There are \(2n\) real numbers around a circle, and their total sum is positive. For each number, consider the two arcs of length \(n\) for which this number is an endpoint. Prove that there is a number for which both such arc sums are positive.
Hint 1. Denote the sum of \(n\) consecutive numbers by \(S_i\).
Hint 2. Use \(S_i+S_{i+n}=S\), where \(S\) is the total sum.
Let the numbers be \(a_1,\ldots,a_{2n}\), with indices modulo \(2n\), and let \(S_i=a_i+a_{i+1}+\cdots+a_{i+n-1}\). The total sum is \(S>0\), so \(S_i+S_{i+n}=S\). Hence some \(S_i\) is positive. If all \(S_i>0\), any number works. Otherwise, while moving around the circle, there is a transition \(S_i>0\), \(S_{i+1}\le0\). Then the length-\(n\) arc ending at \(a_i\) from the other side has sum \(S-S_{i+1}>0\). Thus \(a_i\) is an endpoint of two arcs with positive sums.
A. Source analysis. Main objects: inequalities, order, an extremal element, or an invariant. The obvious first move usually gives only a local estimate. The hidden observation is to choose the right nondecreasing quantity, or to add/multiply inequalities only after signs are controlled. The needed step is an ordering, an invariant, a product transformation, or a boundary case.
F. Difficulty justification. Final level 8: the hidden idea is to pass from numbers to arc sums and find a sign change.
G. Why this is not a one-step exercise. A positive total sum does not directly identify the number; a cyclic transition is needed.