EGMO 2026 Problem 6

Let $p$ be a prime number and let $n$ be a positive integer such that $p$ does not divide $n$. Denote by $k$ the number of positive divisors of $n$, and by $1 = d_1 < d_2 < \cdots < d_k = n$ the positive divisors of $n$. For $i = 1, 2, \ldots, k$, let $c_i$ be the number of positive divisors $\ell$ of $d_i^2$ such that $d_i - \ell$ is divisible by $p$. Prove that

$$(p-1)(c_1 + c_2 + \cdots + c_k) \geq k^2.$$