examday

/TMUA

Let $a_1,\ldots,a_n$ be real numbers, where $n\geq2$ and $a_1+\cdots+a_n>0$.

A student wants to prove that some cyclic rearrangement of the list has every initial sum strictly positive. Put $S_0=0$ and $S_k=a_1+\cdots+a_k$ for $1\leq k\leq n$.

The student proposes the following proof.

1. Choose an index $j$ from $0,\ldots,n-1$ by a rule still to be specified. Use the list

   $\displaystyle a_{j+1},\ldots,a_n,a_1,\ldots,a_j.$

2. An initial sum ending at $a_k$ before reaching $a_n$ is $S_k-S_j$, where $j<k<n$. It is strictly positive by the choice of $j$.

3. The sum ending at $a_n$ is $S_n-S_j>0$.

4. An initial sum continuing past $a_n$ and ending at $a_k$, where $1\leq k\leq j$, is $E$. It is strictly positive by the choice of $j$ and the fact that $S_n>0$.

Which rule for choosing $j$, together with which expression for $E$, makes this proof valid? In each option, minima and maxima are taken over $S_0,\ldots,S_{n-1}$.