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sum (a^2 + b^2)/2ab + 2(ab + bc + ca)/3 >=5

Source: 2023 Greece JBMO TST p3/ easy version of Shortlist 2022 A6 https://artofproblemsolving.com/community/c6h3099025p28018726

May 17, 2024
algebrainequalities

Problem Statement

Let a,b,a, b, and cc be positive real numbers such that a2+b2+c2=3a^2 + b^2 + c^2 = 3. Prove that a2+b22ab+b2+c22bc+c2+a22ca+2(ab+bc+ca)3≥5\frac{a^2 + b^2}{2ab} + \frac{b^2 + c^2}{2bc} + \frac{c^2 + a^2}{2ca} + \frac{2(ab + bc + ca)}{3} \ge 5 When equality holds?
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