The real numbers $a_1 \leq a_2 \leq \cdots \leq a_{672}$ are given such that $$a_1 + a_2 + \cdots + a_{672} = 2018.$$For any $n \leq 672$, there are $n$ of these numbers with an integer sum. What is the smallest possible value of $a_{672}$?
Source: Bulgaria JBMO TST 2018, Day 2, Problem 4
Tags: inequalities
The real numbers $a_1 \leq a_2 \leq \cdots \leq a_{672}$ are given such that $$a_1 + a_2 + \cdots + a_{672} = 2018.$$For any $n \leq 672$, there are $n$ of these numbers with an integer sum. What is the smallest possible value of $a_{672}$?