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Beautiful inequality

Source: Kyiv City MO 2021 Round 1, Problem 10.4

December 21, 2023
inequalities

Problem Statement

Positive real numbers a,b,ca, b, c satisfy a2+b2+c2+a+b+c=6a^2 + b^2 + c^2 + a + b + c = 6. Prove the following inequality:
2(1a2+1b2+1c2)≥3+1a+1b+1c2(\frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2}) \geq 3 + \frac{1}{a} + \frac{1}{b} + \frac{1}{c}
Proposed by Oleksii Masalitin
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