Let $j$ and $k$ be integers. Prove that the inequality $$\lfloor(j+k)\alpha\rfloor+\lfloor(j+k)\beta\rfloor\ge\lfloor j\alpha\rfloor+\lfloor j\beta\rfloor+\lfloor k(\alpha+\beta)\rfloor$$holds for all real numbers $\alpha,\beta$ if and only if $j=k$.