Problem

Source: Turkey TST 2003 - P6

Tags: combinatorics proposed, combinatorics



For all positive integers $n$, let $p(n)$ be the number of non-decreasing sequences of positive integers such that for each sequence, the sum of all terms of the sequence is equal to $n$. Prove that \[\dfrac{1+p(1)+p(2) + \dots + p(n-1)}{p(n)} \leq \sqrt {2n}.\]