Inequality 68 [W. Berndt]

Let $a,b,c$ be nonnegative real numbers such that $a^2+b^2+c^2=1.$ Prove that

$$\frac{1}{1-a^2}+\frac{1}{1-b^2}+\frac{1}{1-c^2}+\frac{1}{1-bc}+\frac{1}{1-ca}+\frac{1}{1-ab} \ge 9.$$

Proof. Download here. 

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