I want to maximize $$\sum_{i,j \in [n], i \ne j} x_i \times x_j$$ where $\forall i ~~$ $0\le x_i \le 1$ and $x_1 + x_2 + x_3 + \ldots +x_n = 1$
I want to prove that the sum will be maximized when $\forall i ~~$ $x_i = \frac{1}{n}$. I don't know whether this statement is true or not.
Note:- It would be more helpful if you can prove that using shifting of weights from one $x_i$ and $x_j$. Even the proof is not following that method it is fine.
On
If you take the derivative wrt $x_i$ you get, for each $i$
$$ \sum_{j=1}^n x_j-x_i-\lambda=0\implies x_i=1-\lambda$$
where $\lambda$ is the Lagrange multiplier. This means that all the $x_i$ need to take the same value. Adding the $n$ FOC's results in
$$ 1=n(1-\lambda)\implies \lambda=\frac{n-1}{n},$$
and, therefore, $$ x_i=\frac{1}{n}$$
Your summation is $x_1(\sum_{j\ne1}x_j)+x_2(\sum_{j\ne2}x_j)+...+x_n(\sum_{j\ne n}x_j)$ which is $x_1(1-x_1)+...+x_n(1-x_n)=(x_1+...+x_n)-(x_1^2+...+x^2_n)=1-(x_1^2+...+x^2_n)$. To maximize this, we need to minimize $\sum_{i\in[n]}x_i^2$.
We know $\sum_{i=1}^nx_i^2\ge \frac{(\sum_{i=1}^nx_i)^2}n=1/n$.
And indeed this minimum is observed when $ x_1=x_2=...=x_n=1/n$. The maximum value is $1-1/n$.