Can we prove the Arithmetic and Geometric Mean Inequality?

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Prove that, if \(a\) and \(b\) are real, \[\frac{1}{2}(a^2+b^2)\geq ab.\]

Deduce that, if \(a\),\(b\) and \(c\) are real, \[a^2+b^2+c^2\geq ab+bc+ca.\]

Show that \(a+b+c\) is a factor of \(a^3+b^3+c^3-3abc\) and find the other factor of this expression.

Deduce that, if \(p\), \(q\) and \(r\) are positive, \[\frac{1}{3}(p+q+r)\geq (pqr)^{\frac{1}{3}}.\]