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Mathematics
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12
If $ab+bc+ca=1,$ prove $\frac{\sqrt{a+1}}{a+bc}+\frac{\sqrt{b+1}}{b+ca}+\frac{\sqrt{c+1}}{c+ab}\ge 1+2\sqrt{2}.$
Oct 8, 2023
11
If $a,b,c\ge 0: ab+bc+ca=1$, find Min $P=\frac{1}{\sqrt{2a+bc}}+\frac{1}{\sqrt{2b+ca}}+\frac{1}{\sqrt{2c+ab}}$
May 20, 2023
10
$a+b+c+abc=4$ and prove $\sum_{cyc}\sqrt{\frac{ab+9}{ab+9c}}\ge 3$.
Jul 13, 2023
9
Minimize $P=\dfrac{\sqrt{a+1}+\sqrt{b+1}+\sqrt{c+1}}{\sqrt{ab+bc+ca+6}},$ if $a+b+c+abc=4.$
Nov 3, 2023
9
If $ab+bc+ca=1,$ prove that $\frac{\sqrt{5a+4}}{a+bc}+\frac{\sqrt{5b+4}}{b+ca}+\frac{\sqrt{5c+4}}{c+ab}\ge 8.$
Oct 20, 2023
9
Prove $\sum\limits_{\mathrm{cyc}} \sqrt{5a+5b+8ab}\ge 3\sqrt{2}+2\sqrt{5}$ for $ab+bc+ca=1$
Jul 28, 2023
8
If $a,b,c\ge 0: ab+bc+ca=3,$ prove $\sqrt{\frac{a}{a+b+6}}+\sqrt{\frac{b}{c+b+6}}+\sqrt{\frac{c}{a+c+6}}\le \frac{3\sqrt{2}}{4}.$
Jul 30, 2023
7
Prove $\sqrt{\frac{a+4bc}{4a+bc}}+\sqrt{\frac{b+4ca}{4b+ac}}+\sqrt{\frac{c+4ab}{4c+ab}}\ge 3,$ when $a+b+c=3.$
Oct 28, 2023
6
How to prove: $ab+bc+ca+\sqrt[3]{abc}+20\ge 2\sum_{cyc}\sqrt{a(4a+ab+bc+ca)}$ if $a+b+c=5$
May 16, 2023
6
Prove: $\sum\frac{\sqrt{a+3}}{a+\sqrt{bc}}\ge\frac{2(\sqrt{a}+\sqrt{b}+\sqrt{c})}{\sqrt{a+b+c+1}}$
Aug 5, 2023