Question 32
Let $a, b, c$ be positive real numbers such that
$$
a+b+c=\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}}
$$
Prove that
$$
2(a+b+c) \geq \sqrt[3]{7 a^{2} b+1}+\sqrt[3]{7 b^{2} c+1}+\sqrt[3]{7 c^{2} a+1}
$$
Find all triples $(a, b, c)$ for which equality holds.
$$
a+b+c=\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}}
$$
Prove that
$$
2(a+b+c) \geq \sqrt[3]{7 a^{2} b+1}+\sqrt[3]{7 b^{2} c+1}+\sqrt[3]{7 c^{2} a+1}
$$
Find all triples $(a, b, c)$ for which equality holds.