KöMaL 2023 A847

Let $A$ be a given finite set with some of its subsets called pretty. Let a subset be called small, if it's a subset of a pretty set. Let a subset be called big, if it has a pretty subset. (A set can be small and big simultaneously, and a set can be neither small nor big.) Let $a$ denote the number of elements of $A$, and let $p$, $s$ and $b$ denote the number of pretty, small and big sets, respectively. Prove that $2^a\cdot p\le s \cdot b$.