Maybe my question is easy but I need some help since I do not see how to answer it.
I have a set of $n$ nonnegative integers $b_i$ with $i=1,...,n$ and a set of rational $x_i$ in the interval $[0,1]$ for all $i=1,...,n$.
Is it always possible to find a set of $n$ nonnegative integers $y_i$ with $i=1,...,n$ such as
$$\sum_{i=1}^nx_iy_i=\left\lceil\sum_{i=1}^nx_ib_i\right\rceil-\sum_{i=1}^nx_ib_i?$$
Example with $n=2$, $b=(26,10)$ and $x=(\frac{1}{5},\frac{1}{2})$: the relation holds with $y=(4,0)$.
Thank you very much for your help!