Question

Prove that \(\lfloor 2x \rfloor + \lfloor 2y \rfloor \geq \lfloor x \rfloor + \lfloor y \rfloor + \lfloor x+y \rfloor\) for all real \(x\) and \(y\). thanks in advance

Answer 1

we either have \(x\le\lfloor x\rfloor+1/2\) or \(x>\lfloor x\rfloor +1/2\), likewise for \(y\). we can break it into a four cases, and then show two (since the other two follow by symmetry)