Prove AM-GM Inequality: a,b,c ≥ 0 and a+b+c=3

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The discussion revolves around proving the AM-GM inequality for non-negative variables a, b, and c, given that their sum equals 3. Participants suggest rewriting the left-hand side of the inequality using the square of the sum of a, b, and c. This leads to the conclusion that ab + ac + bc must be less than or equal to 3. The conversation emphasizes the importance of manipulating the expressions to establish the inequality. Ultimately, the participants express gratitude for the assistance provided in solving the problem.
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Can't figure this out and hope to get some help, TIA!

a,b,c >= 0 and a+b+c=3
Prove that a²+b²+c²+ab+bc+ca >= 6
 
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Write the left hand side of the inequality differently, using the expression for (a+b+c)^2.

You will arrive upon the inequality ab+bc+ac <= 3 = a+b+c. Now, can you prove ab+bc+ac <= (a+b+c)(a+b+c)/3 ?
 
disregardthat <---

Yes, thanks alot
 

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