Let $n\geq 2$ and $x\_1,\ldots,x\_n > 0$ be such that $x\_1+\cdots+x\_n =1$. Is it true that there must exist a positive integer $k$ such that $$\{x\_1k\}+\cdots+\{x\_nk\} = n-1?$$


This looks closely related to the [density of the fractional part](https://math.stackexchange.com/questions/903142/for-an-irrational-number-a-the-fractional-part-of-na-for-n-in-mathbb-n-is). 
Note that the quantity $\{x\_1k\}+\cdots+\{x\_nk\}$ is always an integer, since it equals $k-\lfloor x\_1k\rfloor - \dots - \lfloor x\_nk\rfloor$. Also, as each term is strictly less than one, $n-1$ is the highest value the sum can take.



High sum of fractional parts
If the $x\_i$ are rational, take the lcm of the denominators and decrease it by $1$.


If they are not necessarily rational, act similarly: using, e.g., [Kronecker's theorem](https://en.wikipedia.org/wiki/Kronecker%27s_theorem), take a $k$ such that all the $kx\_i$ are sufficiently close to integers, and decrease that $k$ by $1$.


