$\left\lfloor \frac{a-b}{2} \right\rfloor + \left\lceil \frac{a+b}{2} \right\rceil = a$ when $a,b$ are integers? Let $a$ and $b$ be positive integers.
If $b$ is even, then we have $$\left\lfloor \frac{a-b}{2} \right\rfloor + \left\lceil \frac{a+b}{2} \right\rceil = a$$
I think the equality also hold when $b$ is odd. What could be a proof for it?

As $s:=a+b$ and $a-b$ have the same parity, we can write
$$\left\lfloor\frac{s-2b}2\right\rfloor+\left\lceil\frac s2\right\rceil=\left\lfloor\frac s2\right\rfloor+\left\lceil\frac s2\right\rceil-b=s-b=a.$$
