query: Prove that $(\frac{bc+ac+ab}{a+b+c})^{a+b+c} \geq \sqrt{(bc)^a(ac)^b(ab)^c}$
positive_passages.docid: 148744
positive_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: 41558
negative_passages.text: Prove that $\frac4{abcd} \geq \frac a b + \frac bc + \frac cd +\frac d a$
negative_passages.docid: 99692
negative_passages.text: Prove that $a^3+b^3+c^3 \geq a^2b+b^2c+c^2a$
negative_passages.docid: 18606
negative_passages.text: How prove $\frac{a^2}{a+2b^2}+\frac{b^2}{b+2c^2}+\frac{c^2}{c+2a^2}\ge 1$
negative_passages.docid: 156513
negative_passages.text: Prove that inequality $\sum _{cyc}\frac{a^2+b^2}{a+b}\le \frac{3\left(a^2+b^2+c^2\right)}{a+b+c}$
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: 112969
negative_passages.text: Show that $ab + bc + ac \ge a\sqrt{bc} + b\sqrt{ac} + c\sqrt{ab}$
negative_passages.docid: 14221
negative_passages.text: How can we prove it? ${a}{b}≤\frac{a^p}{p}+\frac{b^q}{q}$
negative_passages.docid: 95852
negative_passages.text: Prove that $(a+b)(b+c)(c+a) \ge8$
negative_passages.docid: 63958
negative_passages.text: how to prove this inequality $(ab+bc+ac)^2 ≥ 3abc(a+b+c)$
negative_passages.docid: 39779
negative_passages.text: If $a,b,c>0$ and $a+b+c=1$, prove that $\sqrt{\frac{ab}{c+ab}}+\sqrt{\frac{bc}{a+bc}}+\sqrt{\frac{ac}{b+ac}} \le \frac{3}{2}$.
negative_passages.docid: 38448
negative_passages.text: Show that a simple inequality holds: $a<b$ implies $\frac ab < \frac{a+1}{b+1}$
negative_passages.docid: 105164
negative_passages.text: If $abc=1$, prove that $\frac{2}{(a+1)^2+b^2+1} + \frac{2}{(b+1)^2+c^2+1} + \frac{2}{(c+1)^2+a^2+1} \le 1$.
negative_passages.docid: 133382
negative_passages.text: Proof that $a+b+\frac{1}{1+\sqrt{ab}}\geq\frac{5}{2}$
negative_passages.docid: 71299
negative_passages.text: Prove that $\frac{a}{b} < \frac{a+c}{b+d} <\frac{ c}{d}$
negative_passages.docid: 5465
negative_passages.text: This inequality $a+b^2+c^3+d^4\ge \frac{1}{a}+\frac{1}{b^2}+\frac{1}{c^3}+\frac{1}{d^4}$
negative_passages.docid: 56101
negative_passages.text: Prove the inequality $\frac{a}{c+a-b}+\frac{b}{a+b-c}+\frac{c}{b+c-a}\ge{3}$