MEMO 2011 pojedinačno problem 2

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28. travnja 2012.
Let n \geq 3 be an integer. John and Mary play the following game: First John labels the sides of a regular n-gon with the numbers 1, 2,\ldots, n in whatever order he wants, using each number exactly once. Then Mary divides this n-gon into triangles by drawing n-3 diagonals which do not intersect each other inside the n-gon. All these diagonals are labeled with number 1. Into each of the triangles the product of the numbers on its sides is written. Let S be the sum of those n - 2 products.

Determine the value of S if Mary wants the number S to be as small as possible and John wants S to be as large as possible and if they both make the best possible choices.
Izvor: Srednjoeuropska matematička olimpijada 2011, pojedinačno natjecanje, problem 2