# TA’s Challenge Problem #3

• Jun 9th 2009, 02:18 AM
TheAbstractionist
TA’s Challenge Problem #3
If $x,y,z$ are non-negative real numbers, show that

$(x+y+z)^2+xy+yz+zx\ \geqslant\ 2\sqrt{xy}(x+y)+2\sqrt{yz}(y+z)+2\sqrt{zx}(z+x)$
• Jun 9th 2009, 03:36 PM
NonCommAlg
Quote:

Originally Posted by TheAbstractionist
If $x,y,z$ are non-negative real numbers, show that

$(x+y+z)^2+xy+yz+zx\ \geqslant\ 2\sqrt{xy}(x+y)+2\sqrt{yz}(y+z)+2\sqrt{zx}(z+x)$

we have $(x+y-2\sqrt{xy})^2 \geq 0,$ which gives us: $x^2 + y^2+6xy \geq 4\sqrt{xy}(x+y).$ similarly: $y^2 + z^2+6yz \geq 4\sqrt{yz}(y+z)$ and $x^2 + z^2+6xz \geq 4\sqrt{xz}(x+z).$

add these three inequalities together and then divide by 2 to get the result.
• Jun 9th 2009, 11:54 PM
TheAbstractionist
Very good! (Clapping)