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I normally consider myself fairly decent with algebra, but when it comes to inequalities... Well, I cannot even solve this very simple one, so please help me!

I have been (somewhat) studying inequalities for the past few days, but I cannot find a theorem that deals with changing from quadratics to quartics (by this I mean two variables multiplied together and four variables multiplied together -- not the traditional x^2 or x^4) as is required by this problem:

Show that for positive a, b, c, and c, such that abcd=1, a^2 +b^2+c^2+d^2 + ab+ac+ad+bc+bd+cd is not smaller then 10.

I think it's fairly clear we need to prove:

a^2 +b^2+c^2+d^2 + ab+ac+ad+bc+bd+cd >= 10abcd=10

If you use any theorems, please mention what they are called. If you derive everything from scratch, all the better!

Thank you for any help or hints you can give me!

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# Traditional x^2 or x^4

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