Question 20
A6 (SWE). For $n\ge3$ and $a_1\le a_2\le\cdots\le a_n$ given real numbers we have the following instructions:
(1) place the numbers in some order in a circle;
(2) delete one of the numbers from the circle;
(3) if just two numbers are remaining in the circle, let $S$ be the sum of these two numbers. Otherwise, if there are more than two numbers in the circle, replace $(x_1,x_2,x_3,\ldots,x_{p-1},x_p)$ with $(x_1+x_2,x_2+x_3,\ldots,x_{p-1}+x_p,x_p+x_1)$. Afterwards, start again with step (2).
Show that the largest sum $S$ that can result in this way is given by the formula
$$S_{\max}=\sum_{k=2}^n\binom{n-2}{\lfloor k/2\rfloor-1}a_k.$$
(1) place the numbers in some order in a circle;
(2) delete one of the numbers from the circle;
(3) if just two numbers are remaining in the circle, let $S$ be the sum of these two numbers. Otherwise, if there are more than two numbers in the circle, replace $(x_1,x_2,x_3,\ldots,x_{p-1},x_p)$ with $(x_1+x_2,x_2+x_3,\ldots,x_{p-1}+x_p,x_p+x_1)$. Afterwards, start again with step (2).
Show that the largest sum $S$ that can result in this way is given by the formula
$$S_{\max}=\sum_{k=2}^n\binom{n-2}{\lfloor k/2\rfloor-1}a_k.$$