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I am stuck with these two floor function problems.

Please help me

1. Let r be a real number, and n be a positive integer.
Prove $[r]+[r+\frac1n]+...+[r+\frac{n-1}n]=[nr]$

2. Let S be set of integers given by $[\alpha x]$ and $[\beta x]$ for x=1,2,3,...
Prove that S consists of every integer, each appearing exactly once, iff $\alpha$ and $\beta$ are positive irrational numbers such that $\frac1\alpha+\frac1\beta=1$

Thank you

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2 is betty's theorem. –  Calvin Lin Oct 18 '13 at 17:15

1 Answer 1

Part 1 hint:

Consider the value of $r-[r]$.

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I tried to a state and not move on. $[\{r\}+\frac1n]+[\{r\}+\frac2n]+...+[\{r\}+\frac{n-1}n]=[n\{r\}]$ where {r}=r-[r]. How can I move to the next step? –  SimplyComplex Oct 18 '13 at 17:18
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@SimplyComplex, $\dfrac{k}{n} \le \{r\} < \dfrac{k+1}{n}$ for some $k \in \mathbb{Z}$ with $0 \le k \le n-1$. –  njguliyev Oct 18 '13 at 17:32

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