2015 USAMO 第 6 题

Consider $0<\lambda<1$, 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 set $A_n$ 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 $\frac{n(n+1)}{2}\lambda$. (A multiset is a set-like collection of elements in which order is ignored, but repetition of elements is allowed and multiplicity of elements is significant. For example, multisets $\{1, 2, 3\}$ and $\{2, 1, 3\}$ are equivalent, but $\{1, 1, 2, 3\}$ and $\{1, 2, 3\}$ differ.)

考虑 $0<\lambda<1$，并令 $A$ 为正整数的多重集。设$A_n=\{a\in A: a\leq n\}$。假设对于每个 $n\in\mathbb{N}$，集合 $A_n$ 最多包含 $n\lambda$ 个数字。证明存在无限多个 $n\in\mathbb{N}$，其中 $A_n$ 中的元素之和至多为 $\frac{n(n+1)}{2}\lambda$。 (多重集是类似集合的元素集合，其中顺序被忽略，但允许元素重复，并且元素的重数很重要。例如，多重集 $\{1, 2, 3\}$ 和 $\{2, 1, 3\}$ 是等价的，但 $\{1, 1, 2, 3\}$ 和 $\{1, 2, 3\}$ 不同。)