Question 128
Let $n \ge 2$ be an integer. We consider two finite subsets $A, B$ of the set of integers, such that the subset $A$ contains at most $n$ elements and let $\Gamma$ be a subset of the set $\{(\alpha, \beta): \alpha \in A, \beta \in B\}$. Achilles writes on a blackboard all possible differences $\alpha - \beta$, with $(\alpha, \beta) \in \Gamma$. Let $d$ be the number of all these differences. Next, Achilles writes in a second blackboard all triads $(\kappa, \lambda, \mu)$ with $(\kappa, \lambda) \in \Gamma, (\kappa, \mu) \in \Gamma$. Let $p$ be the number of all these triads. Prove that:
$$
n \cdot p \ge d^2.
$$
$$
n \cdot p \ge d^2.
$$