<s>[INST] Prove that for all real numbers $x,y$ and $z$ that $x^2+y^2+z^2 \geq x\sqrt{y^2+z^2}+y\sqrt{x^2+z^2}$ 
Prove that for all real numbers $x,y$ and $z$ that $x^2+y^2+z^2 \geq x\sqrt{y^2+z^2}+y\sqrt{x^2+z^2}$.

It looks like AM-GM should be used here but the square roots make it difficult. So maybe Cauchy-Schwarz works?
 [/INST] Note that (by AM-GM),

 $\frac{x^2+(y^2+z^2)}{2}\geq\sqrt{x^2(y^2+z^2)}=x\sqrt{y^2+z^2}$.

and similarly,

 $\frac{y^2+(x^2+z^2)}{2}\geq\sqrt{y^2(x^2+z^2)}=y\sqrt{x^2+z^2}$.

Summing them up gives:

 $x^2+y^2+z^2\geq x\sqrt{y^2+z^2}+y\sqrt{x^2+z^2}$

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