query: How to prove the inequality $\sum_{cyc}\frac{x}{1-x^2}\ge \frac{3\sqrt{3}}{2}$ when $xy+yz+zx=1$?
positive_passages.docid: 150581
positive_passages.text: How do I show that $\frac a{1 - a^2} + \frac b{1 - b^2} + \frac c{1 - c^2} \ge \frac {3 \sqrt 3}2$
negative_passages.docid: 13902
negative_passages.text: Show that $\frac{1}{x_1^2}+\frac{1}{4\cdot x_2^2}+\ldots+\frac{1}{n^2\cdot x_n^2}\leq\frac{3n-2}{2n-1}$.
negative_passages.docid: 138099
negative_passages.text: A cyclic inequality $\sum\limits_{cyc}{\sqrt{3x+\frac{1}{y}}}\geqslant 6$
negative_passages.docid: 58969
negative_passages.text: A proof for the inequality $\sum_{i=0}^{t-2}{\frac{1}{t+3i}} \leq \frac{1}{2} $ for all $t \geq 2$
negative_passages.docid: 92300
negative_passages.text: For positive $a$, $b$, $c$ with $abc=1$, show that $\sum_{cyc}\sqrt{a^2-a+1}\geq a+b+c$
negative_passages.docid: 86209
negative_passages.text: Show this inequality $\frac{n}{a_1 - a_0} + \frac{n - 1}{a_2 - a_1} + \cdots + \frac{1}{a_n - a_{n-1}} \ge \sum_{k=1}^n \frac{k^2}{a_k}$
negative_passages.docid: 60673
negative_passages.text: To prove $\sum_{cyc}\frac{1}{a^3+b^3+abc} \le \frac{1}{abc}$
negative_passages.docid: 117046
negative_passages.text: Show that $\frac{a^2}{\sqrt{(1+a^3)(1+b^3)}}+\frac{b^2}{\sqrt{(1+b^3)(1+c^3)}}+\frac{c^2}{\sqrt{(1+c^3)(1+a^3)}} \geq \frac{4}{3}$
negative_passages.docid: 145783
negative_passages.text: How can I prove this inequality in a triangle: $\frac{1}{(b+c-a)} +\frac{1}{(c+a-b)} +\frac{1}{(a+b-c)} \gt \frac{1}{a} +\frac{1}{b} +\frac{1}{c}$?
negative_passages.docid: 19896
negative_passages.text: Prove $\frac{a}{b}+ \frac{b}{c}+ \frac{c}{a}\geq \frac{1}{a}+ \frac{1}{b}+ \frac{1}{c}$
negative_passages.docid: 157416
negative_passages.text: Prove inequality: $\sum \frac{a^4}{a^3+b^3} \ge \frac{a+b+c}{2}$
negative_passages.docid: 45918
negative_passages.text: Inequality: $x^2+y^2+xy\ge 0$
negative_passages.docid: 66707
negative_passages.text: How to prove this inequality? $ | z-1 | \le | | z | -1 | + | z| \cdot | \arg z | $
negative_passages.docid: 28499
negative_passages.text: Inequality.$\sqrt{\frac{2a}{b+c}}+\sqrt{\frac{2b}{c+a}}+\sqrt{\frac{2c}{a+b}} \leq 3$
negative_passages.docid: 103851
negative_passages.text: Show that $\frac{xy}{z} + \frac{xz}{y} + \frac{yz}{x} \geq x+y+z $ by considering homogeneity
negative_passages.docid: 18313
negative_passages.text: show this inequality with $xy+yz+zx=3$
negative_passages.docid: 87411
negative_passages.text: Prove that $\sqrt{x_1}+\sqrt{x_2}+\cdots+\sqrt{x_n} \geq (n-1) \left (\frac{1}{\sqrt{x_1}}+\frac{1}{\sqrt{x_2}}+\cdots+\frac{1}{\sqrt{x_n}} \right )$