5

Part of 2009 Middle European Mathematical Olympiad

Problems(1)

x^4+y^4+z^4+16(x^2+y^2+z^2) \ge 8(x^3+y^3+z^3)+27

Source: MEMO 2009, problem 1, team competition

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be real numbers satisfying x^2\plus{}y^2\plus{}z^2\plus{}9\equal{}4(x\plus{}y\plus{}z). Prove that x^4\plus{}y^4\plus{}z^4\plus{}16(x^2\plus{}y^2\plus{}z^2) \ge 8(x^3\plus{}y^3\plus{}z^3)\plus{}27 and determine when equality holds.

inequalitiessearchinequalities proposed