MathDB

Let

n

be a positive integer. Let

S=\{a_1,\ldots, a_{k}\}

be a finite collection of at least

n

not necessarily distinct positive real numbers. Let

f(S)=\left(\sum_{i=1}^{k} a_{i}\right)^{n}

and

g(S)=\sum_{1\leq i_{1}<\ldots<i_{n} \leq k} a_{i_{1}}\cdot\ldots\cdot a_{i_{n}}.

Determine

\sup_{S} \frac{g(S)}{f(S)}

Problem Statement