91Ó°ÊÓ

The question asks to show that there is a permutation $i_1,…,i_n$such that $\underset{j=1}{\overset{n}{∑}}{a}_{i,}{a}_{i,+1}<0$where $a_1,…,a_n$are not all equal to $0$and $\underset{i=1}{\overset{n}{∑}}{a}_i=0$.

The question asks to show that there is a permutation $i_1,…,i_n$such that $\underset{j=1}{\overset{n}{∑}}{a}_{i,j}{a}_{i,+1}<0$where $a_1,…,a_n$are not all equal to $0$and $\underset{i=1}{\overset{n}{∑}}{a}_i=0$

Let $I_1,…,I_n$be a random permutation that is equally likely to be any of the $n$ ! Permutations Then:
$E\left[a_{l_j}{a}_{l_{j+1}}\right]=\underset{k}{∑}E\left[a_{l_j}{a}_{l_{j+1}}âˆ\pounds I_j=k\right]P\left\{I_j=k\right\}$

$=\frac{1}{n}\underset{k}{∑}{a}_k·E\left[a_{l_{j+1}}âˆ\pounds I_j=k\right]$
$=\frac{1}{n}\underset{k}{∑}{a}_k·\underset{i}{∑}{a}_i·P\left\{I_{j+1}=iâˆ\pounds I_j=k\right\}$

$=\frac{1}{n·(n-1)}\underset{k}{∑}{a}_k\underset{i=k}{∑}{a}_i$
$=\frac{1}{n·(n-1)}\underset{k}{∑}{a}_k\left(-a_k\right)$

Where the final equality followed from the assumption that $\underset{i=1}{\overset{n}{∑}}{a}_i=0$. Since the preceding shows that
$E\left[\underset{j=1}{\overset{n-1}{∑}}{a}_{l_j}{a}_{l_{j+1}}\right]<0$

Therefore, one can conclude that there must be some permutation $i_1,…,i_n$for which
$$$\underset{j=1}{\overset{n-1}{∑}}{a}_{ij}{a}_{ij+1}<0$