Solution
How Many Variables Can Repel Equally
Show the problem again
Random variables $X_1, \dots, X_n$ each have variance 1, and every pair has correlation exactly $-3/20$. What is the largest $n$ for which such variables can exist?
Worked solution
Look at the variance of the sum: $\operatorname{Var}(\sum X_i) = n + n(n-1)(-3/20)$, which must be non-negative. This gives $1 - 3(n-1)/20 \ge 0$, i.e. $n - 1 \le 20/3$, so $n \le 7.67$ and $n \le 7$. For the equicorrelation matrix the eigenvalues are $1 + (n-1)\rho$ (once) and $1-\rho$ (repeated), so this condition is also sufficient. The largest $n$ is **7**.
Source: Classic positive-semidefiniteness argument; no single originator. Statement written for AxiomIQ.