Let $a,b,c$ be positive real numbers with $abc=1$. Prove the inequality \[\frac{1}{\sqrt{a+2b+6}}+\frac{1}{\sqrt{b+2c+6}}+\frac{1}{\sqrt{c+2a+6}}\le 1.\]

Tintarn wrote: Let $a,b,c,d$ be positive real numbers with $abcd=1$. Prove the inequality \[\frac{1}{\sqrt{a+2b+3c+10}}+\frac{1}{\sqrt{b+2c+3d+10}}+\frac{1}{\sqrt{c+2d+3a+10}}+\frac{1}{\sqrt{d+2a+3b+10}} \le 1.\] The following inequality is also true 1/(a+2b+3c+10)+....<=1/4

sqing wrote: Let $a,b,c$ be positive real numbers with $abc=1$. Prove the inequality \[\frac{1}{\sqrt{a+2b+6}}+\frac{1}{\sqrt{b+2c+6}}+\frac{1}{\sqrt{c+2a+6}}\le 1.\] 1/(a+2b+3)+...<=1/2

Let $a,b,c,d$ be positive real numbers with $abcd=1$. Prove the inequality \[\frac{1}{\sqrt{a+2b+3c+10}}+\frac{1}{\sqrt{b+2c+3d+10}}+\frac{1}{\sqrt{c+2d+3a+10}}+\frac{1}{\sqrt{d+2a+3b+10}} $$ $$\leq 2\sqrt{\frac{1}{a+2b+3c+10}+\frac{1}{b+2c+3d+10}+\frac{1}{c+2d+3a+10}+\frac{1}{d+2a+3b+10} }\leq 1.\]Let $a,b,c$ be positive real numbers with $abc=1$. Prove the inequality $$\frac{1}{\sqrt{a+2b+6}}+\frac{1}{\sqrt{b+2c+6}}+\frac{1}{\sqrt{c+2a+6}}$$$$\leq \sqrt{3\left(\frac{1}{a+2b+6}+\frac{1}{b+2c+6}+\frac{1}{c+2a+6}\right) }\leq1.$$

http://www.pdmi.ras.ru/EIMI/2018/Baltic_way/bw18coord.pdf Let $a,b,c$ be positive real numbers with $abc=1$. Prove the inequality $$\frac{1}{\sqrt{a+2b+3}}+\frac{1}{\sqrt{b+2c+3}}+\frac{1}{\sqrt{c+2a+3}}$$$$\leq \sqrt{3\left(\frac{1}{a+2b+3}+\frac{1}{b+2c+3}+\frac{1}{c+2a+3}\right) }\leq \frac{\sqrt{6}}{2}.$$https://artofproblemsolving.com/community/c4h1670607p10626779

sqing wrote: Let $a,b,c,d$ be positive real numbers with $abcd=1$. Prove the inequality \[\frac{1}{\sqrt{a+2b+3c+10}}+\frac{1}{\sqrt{b+2c+3d+10}}+\frac{1}{\sqrt{c+2d+3a+10}}+\frac{1}{\sqrt{d+2a+3b+10}} $$ $$\leq 2\sqrt{\frac{1}{a+2b+3c+10}+\frac{1}{b+2c+3d+10}+\frac{1}{c+2d+3a+10}+\frac{1}{d+2a+3b+10} }\leq 1.\] Let $a,b,c,d$ be positive real numbers with $abcd=1$. Prove the inequality$$\frac{1}{a^4+2b^4+3c^4+10}+\frac{1}{b^4+2c^4+3d^4+10}+\frac{1}{c^4+2d^4+3a^4+10}+\frac{1}{d^4+2a^4+3b^4+10} \leq 1.$$Proof of dragonheart6:

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sqing wrote: Let $a,b,c$ be positive real numbers with $abc=1$. Prove the inequality $$\frac{1}{\sqrt{a+2b+6}}+\frac{1}{\sqrt{b+2c+6}}+\frac{1}{\sqrt{c+2a+6}}$$$$\leq \sqrt{3\left(\frac{1}{a+2b+6}+\frac{1}{b+2c+6}+\frac{1}{c+2a+6}\right) }\leq1.$$ Let $a,b,c$ be positive real numbers with $abc=1$. Prove the inequality $$\frac{1}{a^3+2b^3+6}+\frac{1}{b^3+2c^3+6}+\frac{1}{c^3+2a^3+6}\leq\frac{1}{3}.$$Proof: $$\frac{1}{a^3+2b^3+6}+\frac{1}{b^3+2c^3+6}+\frac{1}{c^3+2a^3+6}\leq\frac{1}{3}\left(\frac{1}{ab+b+1}+\frac{1}{bc+c+1}+\frac{1}{ca+a+1}\right)=\frac{1}{3}.$$

Let $a,b,c,d$ be positive real numbers such that $abcd=1$. Prove the inequality $$(a^2b+b^2c+c^2d+d^2a)(ab^2+bc^2+cd^2+da^2) \ge(a+c)(b+d)(ac+bd+2).$$

2019 China Mathematical Olympiad Collaborative School Joint Competition： Let $a_1,a_2,\cdots,a_n (n\ge 2)$ be positive real numbers such that $a_1a_2\cdots a_n=1.$ Prove that

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Let $a_1,a_2,\cdots,a_n (n\ge 2)$ be positive real numbers such that $a_1a_2\cdots a_n=1.$ Prove that

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Holy inequality