USAMO 2015 — Days 1 and 2 Question 6

Fix $0 \lt \lambda \lt 1$, and let $A$ be a multiset of positive integers. Let $A_n = \{a\in A: a\leq n\}$. Assume that for every $n \in \mathbb{N}$, the multiset $A_n$ contains at most $n\lambda$ numbers. Show that there are infinitely many $n\in\mathbb{N}$ for which the sum of the elements in $A_n$ is at most $\frac{n(n+1)}{2}\lambda$.

Proposed by Iurie Boreico (credited by Evan Chen).