Algebra · Inequalities · Mean inequalities · Cyclic expressions
Let \(a\), \(b\) and \(c\) be positive real numbers with \(a + b + c = 3\). Prove that
\[ \frac{1}{\sqrt{a^2 + ab + bc}} + \frac{1}{\sqrt{b^2 + bc + ca}} + \frac{1}{\sqrt{c^2 + ca + ab}} \geq \sqrt{3} . \]
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Serbian National Competition (Drzavno takmicenje) 2017, high school grade I, category A, problem 3. Organized by the Mathematical Society of Serbia (DMS). Source