query: Proving that $\lfloor x + k \rfloor=\lfloor x\rfloor + k$
positive_passages.docid: 151012
positive_passages.text: Find the value of $\lfloor x+y \rfloor$ where $x \in \mathbb{R}$, $y \in \mathbb{Z}$
negative_passages.docid: 40973
negative_passages.text: How to find $\lfloor 1/\sqrt{1}+1/\sqrt{2}+\dots+1/\sqrt{100}\rfloor $ without a calculator?
negative_passages.docid: 71394
negative_passages.text: Prove the following floor function identity
negative_passages.docid: 142338
negative_passages.text: Prove that $\lfloor\frac{n+1}{2}\rfloor+\lfloor\frac{n+2}{4}\rfloor+\lfloor\frac{n+4}{8}\rfloor+\lfloor\frac{n+8}{16}\rfloor+ \dots=n$
negative_passages.docid: 81729
negative_passages.text: Combinatorial proof of $\sum_{k=0}^{n}(-1)^{k}\binom{n}{k}(l-k)^n=n!$, using inclusion-exclusion
negative_passages.docid: 13390
negative_passages.text: Recursive Monotone Decreasing Sequence Proof ${x_{k}} = \frac{1}{2}\left({x_{k-1}+\frac{a}{{x_{k-1}}}}\right)$
negative_passages.docid: 65757
negative_passages.text: Prove that $n! + k$ is a composite number
negative_passages.docid: 5934
negative_passages.text: How to prove floor identities?
negative_passages.docid: 114889
negative_passages.text: Solve summation $\sum_{i=1}^n \lfloor e\cdot i \rfloor $
negative_passages.docid: 127983
negative_passages.text: Does $\lim_{x \to 0+} \left(x\lfloor \frac{a}{x} \rfloor\right)=a?$
negative_passages.docid: 17285
negative_passages.text: Proof of greatest integer theorem: floor function is well-defined
negative_passages.docid: 72551
negative_passages.text: Proof that $\rho = \pm1$ implies linear combination
negative_passages.docid: 108704
negative_passages.text: Help with proof of $|x+y| \geq |x|-|y|$
negative_passages.docid: 103347
negative_passages.text: combinatorial proof of summation
negative_passages.docid: 59462
negative_passages.text: Prove that $n!=\prod_{k=1}^n \operatorname{lcm}(1,2,...,\lfloor n/k \rfloor)$ for any $n \in \mathbb N$
negative_passages.docid: 114228
negative_passages.text: Find the number of solutions to $ \lfloor x \rfloor + \lfloor 2x \rfloor + \lfloor 4x \rfloor + \ldots + \lfloor 32x \rfloor =12345$
negative_passages.docid: 140841
negative_passages.text: Induction proof of lower bound for $\sum \sqrt n$