query: prove that if x is a real number, then $\lfloor2x\rfloor=\lfloor x\rfloor+\lfloor x + \frac{1}{2}\rfloor$
positive_passages.docid: 19064
positive_passages.text: Proof about floor function: $\lfloor x\rfloor+\lfloor x + \frac{1}{n} \rfloor+\cdots + \lfloor x + \frac{(n-1)}{n} \rfloor = \lfloor nx \rfloor$
negative_passages.docid: 143306
negative_passages.text: Limit involving floor function: $\lim\limits_{x\to 0} x \left\lfloor\frac1x \right\rfloor$
negative_passages.docid: 27844
negative_passages.text: How do we prove that $\lfloor0.999\cdots\rfloor = \lfloor 1 \rfloor$?
negative_passages.docid: 87567
negative_passages.text: How to prove $1+x \leq e^x~\forall x \in \mathbb{R}?$
negative_passages.docid: 133505
negative_passages.text: Let $f(n)=\sum_{k=0}^{\left\lfloor n/2\right\rfloor} {2k \choose k}{n \choose 2k}$ . Show that $\sum_{n\geq 0}^{} f(n)x^n=\frac{1}{\sqrt{1-2x-3x^2}}$
negative_passages.docid: 5222
negative_passages.text: $\lfloor x\rfloor + \lfloor y\rfloor \leq \lfloor x+y\rfloor$ for every pair of numbers of $x$ and $y$
negative_passages.docid: 16337
negative_passages.text: Find the solution of $\lfloor \sin^{-1}(x)\rfloor>\lfloor \cos^{-1}(x)\rfloor$
negative_passages.docid: 113237
negative_passages.text: Find the sum of the roots of the floor equation
negative_passages.docid: 1449
negative_passages.text: Floor function properties: $[2x] = [x] + [ x + \frac12 ]$ and $[nx] = \sum_{k = 0}^{n - 1} [ x + \frac{k}{n} ] $
negative_passages.docid: 15205
negative_passages.text: Proving that the series 1 + ... + $1 / \sqrt{x}$ < $2 \sqrt{x}$
negative_passages.docid: 1715
negative_passages.text: Proof: How many digits does a number have? $\lfloor \log_{10} n \rfloor +1$
negative_passages.docid: 52029
negative_passages.text: If $x\in\mathbb R$, solve $4x^2-40\lfloor x\rfloor+51=0$.
negative_passages.docid: 3390
negative_passages.text: finding the real values of $x$ such that : $x=\sqrt{2+\sqrt{2-\sqrt{2+x}}}$
negative_passages.docid: 140599
negative_passages.text: Inequality Proof in Real Numbers: If $p+q=1$ then $pq\le\frac14$
negative_passages.docid: 29190
negative_passages.text: $\left\lfloor \frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\frac{1}{\sqrt{4}}+...+\frac{1}{\sqrt{1024}}\right\rfloor =?$
negative_passages.docid: 138316
negative_passages.text: A simple inequality for floor function
negative_passages.docid: 137016
negative_passages.text: Showing that $\lfloor\frac{x-1}3\rfloor=\lfloor\frac{x}3+\frac23\rfloor-1$ and $\lfloor\frac{x+1}3\rfloor=\lfloor\frac{x}3+\frac13\rfloor$.