Find all integers $p>q\ge 0$, such that for all real $x,y \in [0;1]$ following inequality holds $$\lfloor px \rfloor + \lfloor py\rfloor \ge \lfloor qx+y\rfloor+\lfloor x+qy \rfloor$$
I used $x\le [x]<x+1; x=[x]+{x}$ and substituted into the inequality $1) x=y=1;$ $2)x=1;y=0$; $3)x=0,y=1$. But I did not receive any of the results.