query: Prove that $\sum\limits_{k=1}^n\lfloor n/k\rfloor+\lfloor \sqrt{n} \rfloor$ is even.
positive_passages.docid: 96791
positive_passages.text: How to show that $\lfloor n/1\rfloor+\lfloor n/2 \rfloor+....+\lfloor n/n\rfloor+\lfloor{\sqrt{n}}\rfloor$ is even?
negative_passages.docid: 141446
negative_passages.text: Proving by induction that $\sum\limits_{k=2}^n \frac1{k^2}\le 1$
negative_passages.docid: 30414
negative_passages.text: Prove using combinatorics $\sum\limits_{k=0}^n 2^k \binom{n}{k} \binom{n-k}{\lfloor{\frac{n-k}{2}}\rfloor}=\binom{2n+1}{n}$
negative_passages.docid: 30672
negative_passages.text: Show that $\sum\limits_{k=0}^n\binom{2n}{2k}^{\!2}-\sum\limits_{k=0}^{n-1}\binom{2n}{2k+1}^{\!2}=(-1)^n\binom{2n}{n}$
negative_passages.docid: 14197
negative_passages.text: show that $\sum_{i=0}^l \dfrac{{l\choose i}}{{n\choose k + i}} = \dfrac{n+1}{(n-l+1) {n-l\choose k}}$
negative_passages.docid: 27844
negative_passages.text: How do we prove that $\lfloor0.999\cdots\rfloor = \lfloor 1 \rfloor$?
negative_passages.docid: 41974
negative_passages.text: Limit of Series with differences of Floor function
negative_passages.docid: 146023
negative_passages.text: How to prove that $x^2 - \lfloor x\rfloor^2$ is onto?
negative_passages.docid: 129680
negative_passages.text: Prove that $n$ is even.
negative_passages.docid: 99483
negative_passages.text: If $T(n+1)=T(n)+\lfloor \sqrt{n+1}, \rfloor$ $\forall n\geq 1$, what is $T(m^2)$?
negative_passages.docid: 17375
negative_passages.text: When is the limit of a sum equal to the sum of limits?
negative_passages.docid: 102531
negative_passages.text: Prove $\sum\limits^m_{k=0} \frac{2n-k\choose k}{2n-k\choose n}\frac{2n-4k+1}{2n-2k+1}2^{n-2k}=\frac{n\choose m}{2n-2m\choose n-m}2^{n-2m}$ for-
negative_passages.docid: 1525
negative_passages.text: If for any $k$, $\sum\limits_{n=0}^\infty a_n^k=\sum\limits_{n=0}^\infty b_n^k$, then$(a_n)=(b_{σ(n)}),\ σ \in{\mathfrak S}_{\mathbb N}$
negative_passages.docid: 17285
negative_passages.text: Proof of greatest integer theorem: floor function is well-defined
negative_passages.docid: 151012
negative_passages.text: Find the value of $\lfloor x+y \rfloor$ where $x \in \mathbb{R}$, $y \in \mathbb{Z}$
negative_passages.docid: 107258
negative_passages.text: Proving If $k \le \left\lfloor \frac{n}{2} \right\rfloor$ then $\binom{n}{k-1} < \binom{n}{k}$
negative_passages.docid: 113891
negative_passages.text: The limit of a sum of the form $a_0 \sqrt n + a_1 \sqrt {n + 1} +\cdots + a_k \sqrt {n + k}$