parmenides51 wrote: Let $a, b, c$ be nonzero real numbers such that $a + b + c = 0$. Determine the maximum possible value of $\frac{a^2b^2c^2}{ (a^2 + ab + b^2)(b^2 + bc + c^2)(c^2 + ca + a^2)}$ . Let $a,b,c$ is nonzero real number that $a+b+c=0.$Then $$\frac{a^2b^2c^2}{(a^2+ab+b^2)(b^2+bc+c^2)(c^2+ca+a^2)}\leq\frac{4}{27}.$$Equality holds when $2a=2b=-c ,...$ Let $a,b$ be positive real number . Prove that $$\frac{a^2b^2(a+b)^2}{(a^2+ab+b^2)^3}\leq\frac{4}{27}.$$Equality holds when $a=b.$

sqing wrote: Let $a,b$ be positive real number . Prove that $$\frac{a^2b^2(a+b)^2}{(a^2+ab+b^2)^3}\leq\frac{4}{27}.$$Equality holds when $a=b.$ Proof of NJOY: By AM-GM, \begin{align*}2(a^2+ab+b^2)&=\frac{a(a+b)}2 + \frac{a(a+b)}2 + \frac{b(a+b)}2 + \frac{b(a+b)}2 + a^2 + b^2 \\& \geq 6 \sqrt[6]{\frac{a^4b^4(a+b)^4}{2^4}} = 6 \sqrt[3]{\frac{a^2b^2(a+b)^2}{2^2}},\\& a^2+ab+b^2\geq 3 \sqrt[3]{\frac{a^2b^2(a+b)^2}{2^2}}\iff \frac{a^2b^2(a+b)^2}{(a^2+ab+b^2)^3}\leq\frac{4}{27}.\end{align*}

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parmenides51 wrote: Let $a, b, c$ be nonzero real numbers such that $a + b + c = 0$. Determine the maximum possible value of $\frac{a^2b^2c^2}{ (a^2 + ab + b^2)(b^2 + bc + c^2)(c^2 + ca + a^2)}$ .

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sqing wrote: parmenides51 wrote: Let $a, b, c$ be nonzero real numbers such that $a + b + c = 0$. Determine the maximum possible value of $\frac{a^2b^2c^2}{ (a^2 + ab + b^2)(b^2 + bc + c^2)(c^2 + ca + a^2)}$ . Let $a,b,c$ is nonzero real number that $a+b+c=0.$Then $$\frac{a^2b^2c^2}{(a^2+ab+b^2)(b^2+bc+c^2)(c^2+ca+a^2)}\leq\frac{4}{27}.$$Equality holds when $2a=2b=-c ,...$ Let $a,b$ be positive real number . Prove that $$\frac{a^2b^2(a+b)^2}{(a^2+ab+b^2)^3}\leq\frac{4}{27}.$$Equality holds when $a=b.$ https://artofproblemsolving.com/community/c6h554235p3220767