她改变了不等式的面貌(243)

2022-04-03 05:04阅读：

http://blog.sina.cn/dpool/blog/u/1276195074

Let $a,b,c>0$ and $a+2b+c=\frac 1a+\frac 2b+\frac 1c.$ Prove that $\frac 1{a+2}+\frac 1{b+2}+\frac 1{c+2}\leq 2-\frac{2\sqrt 2}{3}$ $\frac 1{a+k}+\frac 1{b+k}+\frac 1{c+k}\leq \frac{3k-2\sqrt 2}{k^2-1}$ Where $2\leq k \in N^+.$
Let $a,b,c$ be positive real numbers . Prove that $\frac {a+b}{c^2}+\frac {b+c}{a^2}+\frac {c+a}{b^2}\geq \frac{\sqrt{13+16\sqrt{2}}-3}{2}\left(\frac{1}{a}+\frac{2}{b}+\frac{1}{c}\right)$ Let $a,b,c$ be positive real numbers . Prove that $\frac {(a + b)(b + c)(a+2b+c)}{abc }\geq\frac {27} {2}$ $\frac {(a + b)(b + c)(a+6b+c)}{abc }\geq\frac {64} {3}$ Let $a,b,c$ be positive real numbers . Prove that
$\frac {(a + b)(b + c)(a+3b+c)}{abc }\geq\frac {47+13\sqrt{13}} {6}$
Let $a,b,c$ be positive real numbers . Prove that $\frac {(a + b)(b + c)(2a+b+2c)}{abc }\geq9+6\sqrt{3}$ $\frac {(a + b)(b + c)(3a+2b+3c)}{abc }\geq\frac {37+14\sqrt{7}} {3}$ Let $a,b,c$ be positive real numbers . Prove that $\frac {(a + b)(b + c)(a+kb+c)}{abc }\geq\frac {2k^2+10k-1+(4k+1)\sqrt{4k+1}} {2k}$ Where $k>0.$
Let $a,b,c>0.$ Prove that $\frac{a}{b}+\sqrt[3]{\frac{b^3}{c^3}+\frac{c^3}{a^3}}\geq\frac{3}{\sqrt[9]{8}}$ $\frac{a}{b}+2\sqrt[3]{\frac{b^3}{c^3}+\frac{c^3}{a^3}}\geq 3\sqrt[9]{4}$ $\frac{a}{b}+4\sqrt{1+\frac{b}{c}}+\sqrt[3]{1+\frac{c}{a}}>7$