ISL 1999 A6

For $n \geq 3$ and $a_{1} \leq a_{2} \leq \ldots \leq a_{n}$ given real numbers we have the following instructions:

• place out the numbers in some order in a ring;
• delete one of the numbers from the ring;
• if just two numbers are remaining in the ring: let $S$ be the sum of these two numbers. Otherwise, if there are more the two numbers in the ring, replace

Afterwards start again with the step (2). Show that the largest sum $S$ which can result in this way is given by the formula

$$S_{max}= \sum^n_{k=2} \begin{pmatrix} n -2 [\frac{k}{2}] - 1\end{pmatrix}a_{k}.$$

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