query: How can I prove that $27\leq \left( x^{2}+2\right) \left( y^{2}+2\right) \left( z^{2}+2\right) \leq 44,\ x+y+z=3$
positive_passages.docid: 147145
positive_passages.text: Interesting short Inequality
negative_passages.docid: 29679
negative_passages.text: Prove: $|x-y|\leq |x|+|y|$
negative_passages.docid: 105047
negative_passages.text: Prove the inequality $|xy|\leq\frac{1}{2}(x^2+y^2)$
negative_passages.docid: 81842
negative_passages.text: How to prove by induction that $\frac{a^n+b^n}{2}\geq\left(\frac{a+b}{2}\right)^n$?
negative_passages.docid: 135365
negative_passages.text: If $x, y, z$ are the side lengths of a triangle, prove that $x^2 + y^2 + z^2 < 2(xy + yz + xz)$
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: 139718
negative_passages.text: Prove $6 \nmid [\left( \sqrt[3]{28} - 3 \right)^{-n}]$
negative_passages.docid: 87907
negative_passages.text: How to find the maximum and minimum value of $\left|(z_1-z_2)^2 + (z_2-z_3)^2 + (z_3-z_1)^2\right|$ (where $|z_1|=|z_2|=|z_3|=1$)?
negative_passages.docid: 80826
negative_passages.text: To prove $\left|\frac{p_n(z)}{q_m(z)}\right|\leq \frac{M}{|z|^{m-n}}$ for some $M>0$
negative_passages.docid: 63958
negative_passages.text: how to prove this inequality $(ab+bc+ac)^2 ≥ 3abc(a+b+c)$
negative_passages.docid: 56796
negative_passages.text: Maximum of $x^3+y^3+z^3$ with $x+y+z=3$
negative_passages.docid: 41383
negative_passages.text: Prove that $(x+y+z)^2(yz+zx+xy)^2 \leq 3(y^2+yz+z^2)(z^2+zx+x^2)(x^2+xy+y^2)$
negative_passages.docid: 145884
negative_passages.text: Showing $\left|\frac{a+b}{2}\right|^p+\left|\frac{a-b}{2}\right|^p\leq\frac{1}{2}|a|^p+\frac{1}{2}|b|^p$
negative_passages.docid: 117281
negative_passages.text: How prove this inequality $x^3y+y^3z+z^3x\ge xyz(x+y+z)$
negative_passages.docid: 136274
negative_passages.text: $n! \leq \left( \frac{n+1}{2} \right)^n$ via induction
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: 137791
negative_passages.text: Proving inequality $\sqrt{\frac{2a}{b+c}}+\sqrt{\frac{2b}{c+a}}+\sqrt{\frac{2c}{a+b}} \leq \sqrt{3 \left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)}$