<s>[INST] $\left\lfloor \frac{a-b}{2} \right\rfloor + \left\lceil \frac{a+b}{2} \right\rceil = a$ 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?
 [/INST] The equality only depends on the parity of $a+b$, as it is the same as that of $a-b$. Then
$$\left\lfloor\frac02\right\rfloor+\left\lceil\frac02\right\rceil=0$$
and
$$\left\lfloor\frac12\right\rfloor+\left\lceil\frac12\right\rceil=1$$
are enough as a proof.
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