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[r] denotes the greatest integer less than or equal to r

{r} denotes the fractional part of r

This problem is tripping me up. Obviously r has to be some sort of decimal, or else the fractional part of it would be 0. The fact that [r] is the floor of r also makes this confusing for me. Overall, this problem just feels hard to work with.

edit: got it! thank you guys!

Matt Mcstoffel
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    Write $r = n+\alpha$ for $\alpha \in [0,1)$. Then $(n+\alpha)n = 2019 \alpha \implies \alpha = \frac{n^2}{2019-n}$. Having $\alpha \ge 0$ and $\alpha < 1$ corresponds to $n < 2019$ and $2019-n < n^2$. Now solve the quadratic. – mathworker21 Jan 19 '19 at 22:36
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    Whenever you have an equation that involves fractional and integer parts, if you don't have any better ideas you should write $r$ as mathworker21 suggests and see where that leads. Often you can get rid of the fractional parts and floors and get to a normal equation. – Ross Millikan Jan 19 '19 at 22:50
  • The FAQ encourages you to write up your own question if you figure it out. After a couple days you can accept the answer as well. That way it doesn't stay unanswered. – Ross Millikan Jan 20 '19 at 02:47

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