query: prove：$\left|x+\frac{1}{x}\right|\geq 2$
positive_passages.docid: 44233
positive_passages.text: How to prove this inequality $ x + \frac{1}{x} \geq 2 $
negative_passages.docid: 56315
negative_passages.text: Prove $\frac{|a+b|}{1+|a+b|}<\frac{|a|}{1+|a|}+\frac{|b|}{1+|b|}$.
negative_passages.docid: 66387
negative_passages.text: Prove $(x_1+.....+x_n)^2 \leq n(x_1^2 + .....+x_n^2)$
negative_passages.docid: 136403
negative_passages.text: Prove that $\sqrt{a}\leq\frac{1+a}{2}$
negative_passages.docid: 136274
negative_passages.text: $n! \leq \left( \frac{n+1}{2} \right)^n$ via induction
negative_passages.docid: 14952
negative_passages.text: Prove: $\frac{x}{\sqrt{y}}+\frac{y}{\sqrt{x}}\geq \sqrt{x}+\sqrt{y}$
negative_passages.docid: 42545
negative_passages.text: If $ a,b,c\in \left(0,\frac{\pi}{2}\right)\;,$ Then prove that $\frac{\sin (a+b+c)}{\sin a+\sin b+\sin c}<1$
negative_passages.docid: 57670
negative_passages.text: Show that $|z_1 + z_2|^2 < (1+C)|z_1|^2 + \left(1 + \frac{1}{C}\right) |z_2|^2$
negative_passages.docid: 99676
negative_passages.text: Prove $\sum{\frac{1}{(x+2y)^2}} \geq\frac{1}{xy+yz+zx}$
negative_passages.docid: 99753
negative_passages.text: How to prove $\int_0^1\frac{1-x}{(\ln x)(1+x)}\ dx=\ln\left(\frac2{\pi}\right)$?
negative_passages.docid: 73892
negative_passages.text: Prove $Σ_{cyc}\frac{1}{\left(a+1\right)^2}+\frac{1}{a+b+c+1}\ge 1$
negative_passages.docid: 78174
negative_passages.text: How prove this inequality $\left(\int_{0}^{1}f(x)dx\right)^2\le\frac{1}{12}\int_{0}^{1}|f'(x)|^2dx$
negative_passages.docid: 158153
negative_passages.text: Prove: $ \lim_{n\rightarrow\infty} x ^ {\left(\frac 1 n\right)} = 1 $
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: 40700
negative_passages.text: Proving inequality $(a+\frac{1}{a})^2 + (b+\frac{1}{b})^2 \geq \frac{25}{2}$ for $a+b=1$
negative_passages.docid: 77130
negative_passages.text: How to prove this inequality $\left|x\sin{\frac{1}{x}}-y\sin{\frac{1}{y}}\right|<2\sqrt{|x-y|}$?
negative_passages.docid: 33097
negative_passages.text: Proving an inequality without an integral: $\frac {1}{x+1}\leq \ln (1+x)- \ln (x) \leq \frac {1}{x}$