query: Two last cases of the inequality $\sum\limits_{cyc}\frac{a-b}{b+c}\geq0$
positive_passages.docid: 146885
positive_passages.text: How prove this inequality $\frac{a-b}{b+c}+\frac{b-c}{c+d}+\frac{c-d}{d+e}+\frac{d-e}{e+a}+\frac{e-a}{a+b}\ge 0$
negative_passages.docid: 152053
negative_passages.text: Proof that $\frac{a^2}{b} + \frac{b^2}{c} + \frac{c^2}{a} \geq a+b+c$
negative_passages.docid: 91131
negative_passages.text: Question regarding an inequality
negative_passages.docid: 73771
negative_passages.text: An inequality for a sum of sine functions
negative_passages.docid: 14563
negative_passages.text: Show $\frac{b}{c} + \frac{c}{a} + \frac{a}{b} \ge 3$ for $a,b,c > 0$.
negative_passages.docid: 123264
negative_passages.text: Showing $\sum\limits_{cyc}ab\le\frac14\left(\sum\limits_{cyc}a\right)^2$
negative_passages.docid: 113891
negative_passages.text: The limit of a sum of the form $a_0 \sqrt n + a_1 \sqrt {n + 1} +\cdots + a_k \sqrt {n + k}$
negative_passages.docid: 130553
negative_passages.text: How do I prove the following inequality $\sum_{k=n+1}^{\infty}\frac{1}{k^{2}\log k}\leq\frac{1}{n\log n}$?
negative_passages.docid: 124868
negative_passages.text: Prove that $\frac{bc}{b+c}+\frac{ca}{c+a}+\frac{ab}{a+b}<\frac{a+b+c}{2}$
negative_passages.docid: 90902
negative_passages.text: Prove $\frac{a+b+c}{abc} \leq \frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2}$.
negative_passages.docid: 6613
negative_passages.text: AM-GM inequality: $\frac{a^2}{b} + \frac{b^2}{c} + \frac{c^2}{d} + \frac{d^2}{a} \geq a + b + c + d$
negative_passages.docid: 82612
negative_passages.text: Show that $\frac c {1+c} \le \frac a {1+a} + \frac b {1+b}$ , for $c \le a+b$ and $a,b,c \ge 0$
negative_passages.docid: 126032
negative_passages.text: Proving a certain inequality
negative_passages.docid: 137507
negative_passages.text: triangular series and inequality
negative_passages.docid: 4099
negative_passages.text: Inequality $\sum\limits_{cyc}\frac{a^3}{13a^2+5b^2}\geq\frac{a+b+c}{18}$
negative_passages.docid: 141525
negative_passages.text: Cyclic Inequality $\frac{a^3}{(a+b)(a+c)}$
negative_passages.docid: 134535
negative_passages.text: Inequality $\frac{a}{b} + \frac{b}{c} + \frac{c}{a} \geq a + b+ c$ when $abc = 1$