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Difference between revisions of "2015 USAMO Problems/Problem 6"

(Created page with "===Problem 6=== Consider <math>0<\lambda<1</math>, and let <math>A</math> be a multiset of positive integers. Let <math>A_n=\{a\in A: a\leq n\}</math>. Assume that for every <...")
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Revision as of 14:48, 31 January 2016

Problem 6

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.)

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