Question 101
Let $n$ be a composite positive integer and let $1 = d_1 \lt d_2 \lt d_3 \lt \dots \lt d_k = n$ be the divisors of $n$, where $k \ge 3$. Assume that all the equations $d_{i+2}x^2 - 2d_{i+1}x + d_i = 0$, for $i \in \{1, 2, \dots, k-2\}$ have real solutions. Prove that $n = p^{k-1}$ for some prime number $p$.