query: How to prove $(a_1+a_2+\dots a_n)\left(\frac{1}{a_1}+\frac{1}{a_2}+\dots+\frac{1}{a_n}\right)\ge n^2$?
positive_passages.docid: 2455
positive_passages.text: Prove that $\left(\sum^n_{k=1}x_k\right)\left(\sum^n_{k=1}y_k\right)\geq n^2$
negative_passages.docid: 121177
negative_passages.text: Prove that $\frac{(2a+b+c)^2}{2a^2+(b+c)^2}+\frac{(2b+c+a)^2}{2b^2+(c+a)^2}+\frac{(2c+a+b)^2}{2c^2+(a+b)^2} \le 8$
negative_passages.docid: 104658
negative_passages.text: proving $\csc^2 \left( \frac{\pi}{7}\right)+\csc^2 \left( \frac{2\pi}{7}\right)+\csc^2 \left( \frac{4\pi}{7}\right)=8$
negative_passages.docid: 145423
negative_passages.text: prove that if $1 \le m \le n$ then $\left(1+\frac1n\right)^m < 1+\frac mn + \left(\frac mn\right)^2$
negative_passages.docid: 128415
negative_passages.text: Prove that $1<\frac{1}{1001}+\frac{1}{1002}+\frac{1}{1003}+\dots+\frac{1}{3001}<\frac43$
negative_passages.docid: 67666
negative_passages.text: show that $a^2+b^2+c^2+2abc+3\ge (1+a)(1+b)(1+c)$
negative_passages.docid: 5694
negative_passages.text: How to prove $a^2 + b^2 + c^2 \ge ab + bc + ca$?
negative_passages.docid: 84213
negative_passages.text: How to show $n\sum_{i=1}^n {x_i^2} \ge (\sum_{i=1}^n{x_i})^2$
negative_passages.docid: 45742
negative_passages.text: Prove that ${\left(\frac{x+1}{2}\right)}^{x+1} \leq x^x$
negative_passages.docid: 85626
negative_passages.text: prove $\sum_{i=1}^{n}(x_i-\bar{x})^2\lt\sum_{i=1}^{n}(x_i-a)^2$
negative_passages.docid: 89728
negative_passages.text: Prove that $1<\frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{3n+1}$
negative_passages.docid: 148744
negative_passages.text: Prove that $(\frac{bc+ac+ab}{a+b+c})^{a+b+c} \ge \sqrt{(bc)^a(ac)^b(ab)^c}$
negative_passages.docid: 34767
negative_passages.text: Proving $\left(1+\dfrac{1}{1^3}\right)\left(1+\dfrac{1}{2^3}\right)\cdots\left(1+\dfrac{1}{n^3}\right)<3$ for all positive integers $n$
negative_passages.docid: 36837
negative_passages.text: Prove $\frac{1+a^{2}}{1+b+c^{2}} +\frac{1+b^{2}}{1+c+a^{2}} +\frac{1+c^{2}}{1+a+b^{2}} \geq 2$
negative_passages.docid: 30688
negative_passages.text: Prove that $\frac{x_1}{1+x_2+x_3+\ldots+x_n}+\frac{x_2}{1+x_1+x_3+\ldots+x_n}+\ldots+\frac{x_n}{1+x_1+x_2+\ldots+x_{n-1}}\ge\frac{n}{2n-1}$.
negative_passages.docid: 26876
negative_passages.text: How to prove the inequality $2\sqrt{n + 1} − 2 \le 1 +\frac 1 {\sqrt 2}+\frac 1 {\sqrt 3}+ \dots +\frac 1 {\sqrt n} \le 2\sqrt n − 1$?
negative_passages.docid: 150581
negative_passages.text: How do I show that $\frac a{1 - a^2} + \frac b{1 - b^2} + \frac c{1 - c^2} \ge \frac {3 \sqrt 3}2$