<s>[INST] How do I prove thcd It makes me think of means... The RHS Is like a geometric mean but the "divided by 4" annoyes me. The LHS is probably an arithmetic mean... Can they be combined?
 [/INST] By Cauchy-Schwarz,
$$\sqrt{\frac{a^2 + b^2 + c^2 + d^2}{4}}
=\sqrt{a^2 + b^2 + c^2 + d^2}\cdot \sqrt{4\cdot\frac{1}{4^2}}\geq \frac{a+b+c+d}{4}\\\geq  \sqrt[3]{\frac{abc + abd + acd + bcd}{4}}$$
where in the last step we use 
Inequality. $\frac{1}{16}(a+b+c+d)^3 \geq abc+bcd+cda+dab$
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