query: If $a,b\ge 0$ and $a,b \in \mathbb{R}$, then prove $\frac{a+b}{2}\ge\sqrt{ab}$
positive_passages.docid: 13228
positive_passages.text: Proofs of AM-GM inequality
negative_passages.docid: 157416
negative_passages.text: Prove inequality: $\sum \frac{a^4}{a^3+b^3} \ge \frac{a+b+c}{2}$
negative_passages.docid: 11895
negative_passages.text: If $AB=0$ prove that $\mathrm{rank}(A)+\mathrm{rank}(B)\leq n$
negative_passages.docid: 17289
negative_passages.text: prove $\sqrt{a_n b_n}$ and $\frac{1}{2}(a_n+b_n)$ have same limit
negative_passages.docid: 67666
negative_passages.text: show that $a^2+b^2+c^2+2abc+3\ge (1+a)(1+b)(1+c)$
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: 56315
negative_passages.text: Prove $\frac{|a+b|}{1+|a+b|}<\frac{|a|}{1+|a|}+\frac{|b|}{1+|b|}$.
negative_passages.docid: 88425
negative_passages.text: Prove that $\,\sqrt [n] n < 1 + \sqrt{\frac{2}{n}}\,$
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: 38448
negative_passages.text: Show that a simple inequality holds: $a<b$ implies $\frac ab < \frac{a+1}{b+1}$
negative_passages.docid: 14480
negative_passages.text: Proving $(P(AB))^2+(P(AB^c))^2+(P(A^cB))^2+(P(A^cB^c))^2\ge\frac{1}{4}$ for any events $A$ and $B$
negative_passages.docid: 98454
negative_passages.text: Prove that, $\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}\ge 4$ where we do not use AM-GM inequality on the given statement to prove it.
negative_passages.docid: 34275
negative_passages.text: How do I show that $\frac {a^2}b + \frac {b^2}c + \frac {c^2}d + \frac {d^2}a \ge 4$ for $a^2 + b^2 + c^2 + d^2 = 4$?
negative_passages.docid: 257
negative_passages.text: Prove that : $m_{a}m_{b}m_{c}\leq\frac{Rs^{2}}{2}$
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: 5694
negative_passages.text: How to prove $a^2 + b^2 + c^2 \ge ab + bc + ca$?
negative_passages.docid: 64277
negative_passages.text: How to prove ${{a}^{a}}{{b}^{b}}\ge {{\left(\frac{a+b}{2}\right)}^{a+b}}$ ?thanks.