An APMO problem: arithmetic progression and floor function

[Thanic Nur Samin]

Let $a_1, a_2,\ldots a_k$ and $b_1, b_2,\ldots b_k$ be $2k$ real numbers. For a positive integer $n$, define
\[x_n=\displaystyle{\sum_{i=1}^{k}\left\lfloor a_in+b_i \right\rfloor}\]

If $x_n$ is an arithmetic sequence, then prove that $\displaystyle{\sum_{i=1}^k a_i}$ is a integer.

[Thanic Nur Samin]

The sequence $y_n=\displaystyle{\sum_{i=1}^k a_in+b_i}$ is also an arithmetic progression.