## anonymous one year ago Show that $[x]+[x+\frac{1}{2}]=[2x]$ where $[x]$ is the greatest integer of x

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1. perl

You can consider cases. When x is an integer, when x is not an integer

2. anonymous

3. Empty

Here's a way you can solve this that I just came up with so I don't know if there's an easier way. First let's define $$0 \le d<\frac{1}{2}$$ Then this means we can reach all numbers as either: $$x=n+d$$ or $$x=n+d+\frac{1}{2}$$. Why these weird choices? Cause when we plug them in we know no matter what $$d$$ is we have: $[n+d] = n$ $[n+d+\frac{1}{2}] = n+1$ So we have two separate cases, plugging 'em in to: $[x]+[x+\frac{1}{2}]=[2x]$ I think that will end up working?

4. amoodarya

"empty' give the answer i give another way you can also take two case below and check both of them $\lfloor x \rfloor =2n\\ \lfloor x \rfloor=2n+1$