query: Inequality $\sum_{cyc} \sqrt{\frac{a}{a+8}} \geq 1$
positive_passages.docid: 151154
positive_passages.text: Prove the inequality $\sqrt\frac{a}{a+8} + \sqrt\frac{b}{b+8} +\sqrt\frac{c}{c+8} \geq 1$ with the constraint $abc=1$
negative_passages.docid: 104272
negative_passages.text: Inequality of $\vert \text{tanh}(x)-x+\frac{x^3}{3}\vert\leq\frac{1}{8}$ for $x\in[-1/4,1/4]$
negative_passages.docid: 126557
negative_passages.text: Proving the inequality $\sum_{\text{cyc}} \frac{a}{a^2+b^3+c^3}\le\frac1{5abc}$
negative_passages.docid: 52204
negative_passages.text: Proof of one inequality $a+b+c\leq\frac{a^3}{bc}+\frac{b^3}{ca}+\frac{c^3}{ab}$
negative_passages.docid: 154085
negative_passages.text: $\sum a_i \ln(b_i) \leq \sum a_i \ln(a_i)$ with $\sum a_i = \sum b_i = 1$
negative_passages.docid: 95643
negative_passages.text: Proving $\sum_{cyc} \sqrt{a^2+ab+b^2}\geq\sqrt 3$ when $a+b+c=3$
negative_passages.docid: 90343
negative_passages.text: Proving the inequality $|e^z-1|\leq e^{|z|}-1$
negative_passages.docid: 52625
negative_passages.text: Prove inequality $\sqrt{\frac{2a}{b+a}} + \sqrt{\frac{2b}{c+b}} + \sqrt{\frac{2c}{a+c}} \leq 3$
negative_passages.docid: 120093
negative_passages.text: Proof of inequality $2(\sqrt{n+1}-\sqrt{n}) < \frac{1}{\sqrt{n}} < 2(\sqrt{n} - \sqrt{n-1})$ using induction
negative_passages.docid: 56101
negative_passages.text: Prove the inequality $\frac{a}{c+a-b}+\frac{b}{a+b-c}+\frac{c}{b+c-a}\ge{3}$
negative_passages.docid: 69653
negative_passages.text: Inequality $\left(a-1+\frac{1}b\right)\left(b-1+\frac{1}c\right)\left(c-1+\frac{1}a\right)\leq1$
negative_passages.docid: 133428
negative_passages.text: How prove this inequality $\frac{a}{b+3}+\frac{b}{c+3}+\frac{c}{d+3}+\frac{d}{a+3}\le 1$
negative_passages.docid: 147785
negative_passages.text: Proving inequality $x^xy^y \geq (\frac{x+y}{2})^{x+y}$
negative_passages.docid: 156555
negative_passages.text: Inequality $\sqrt{1+x^2}+\sqrt{1+y^2}+\sqrt{1+z^2} \le \sqrt{2}(x+y+z)$
negative_passages.docid: 118559
negative_passages.text: Prove the inequality $\sum_{1\le i<j\le n}\sqrt{a_ia_j}<\frac{n-1}2\sum_{i=1}^n a_i$
negative_passages.docid: 73771
negative_passages.text: An inequality for a sum of sine functions
negative_passages.docid: 134535
negative_passages.text: Inequality $\frac{a}{b} + \frac{b}{c} + \frac{c}{a} \geq a + b+ c$ when $abc = 1$