Show that if $X_1, X_2,...,X_{11}$ are random variables, and $\operatorname{corr}(X_i,X_j) = r$ for $ i\neq j$ , then show $r \geq -1/10$

Question

$\DeclareMathOperator{\corr}{\mathsf{corr}}$Show that if $X_1, X_2,...,X_{11}$ are random variables, and $\corr(X_i,X_j) = r, i\neq j$ , then show $r \geq -1/10$.

I think I need to do this with induction for $n$, so if $X_1, X_2,...,X_{n}$ are random variables, and $\corr(X_i,X_j) = r$ for $i\neq j$ , then $r \geq -1/(n-1)$. I got the first step, when $n = 2$ it's saying if $X_1,X_2$ are random variables, then $\corr(X_1,X_2)\geq -1$, which is true because $|\operatorname{cov}(X_1,X_2)| = |E[(X_1 - E(X_1))(X_2 -E(X_2))]| \leq D(X_1)D(X_2)$, where $D$ is the variance, so $|\corr(X_1,X_2)| \leq 1 \implies \corr(X_1,X_2)\geq -1$.
I can't go any further when I assume it's true for $n-1$.

Answer 1

No need for induction. First, show that $$ \mathbb{V} \left ( \sum_{i=1}^n X_i\right ) = n \sigma^2 + n(n-1) r \sigma^2 $$ then, combine this with the fact that variance is non-negative