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Mathematics
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Anonymous
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8
How to prove $2\left(\sqrt{ab-1}+\sqrt{bc-1}+\sqrt{ca-1}\right)\le \left(\sqrt{a}+\sqrt{b}+\sqrt{c}\right)\sqrt{\sqrt{ab}+\sqrt{bc}+\sqrt{ca}}.$?
Nov 23, 2023
7
Finding $\small{\max\limits_{ab+bc+ca=1}\dfrac{\sqrt{a+bc}+\sqrt{b+ca}+\sqrt{c+ab}}{\sqrt{3a+3b+3c+10}} }$
Oct 30, 2023
7
If $a^2+b^2+c^2+abc=4$, Find minimum $P=\sqrt{\frac{2a+bc}{3}}+\sqrt{\frac{2b+ca}{3}}+\sqrt{\frac{2c+ab}{3}}-\frac{3(a+b+c+abc)}{2}$
Aug 8, 2023
5
How to prove $\sqrt{2a+3bc}+\sqrt{2b+3ca}+\sqrt{2c+3ab}\ge 3\sqrt{5}$?
Nov 28, 2023
5
Prove $2(a+b+c)\left(1+\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)\ge 3(a+b)(b+c)(c+a)$ for $abc=1.$
Nov 27, 2023
5
Prove $\frac{a}{\sqrt{4a+3bc}}+\frac{b}{\sqrt{4b+3ca}}+\frac{c}{\sqrt{4c+3ab}}\le \frac{\sqrt{a+b+c+2}}{2}.$
Nov 2, 2023
3
Proving $\frac{n(n-1)}{2}\le \sum_{k=1}^{n^2}\{\sqrt{k}\}\le \frac{n^2-1}{2}.$
Dec 19, 2023
3
Solving $(a^2+b^2+c^2)\left(\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\right)\ge(3+2\sqrt{3})^2$
Dec 19, 2023
3
Help to prove $\frac{a}{(b^2+3)^2}+\frac{b}{(c^2+3)^2}+\frac{c}{(a^2+3)^2}\ge\frac{3}{16}$ for $a+b+c=3.$
Dec 13, 2023
3
Finding minimum $\sqrt{a(2b+1)}+\sqrt{b(2c+1)}+\sqrt{c(2a+1)}$?
Dec 2, 2023