View source for ASIA TEAM Problems/Problem 6

Dividing both sides of the second equation by <math>xyz</math> gives
<math>\frac{9}{x}+\frac{4}{y}+\frac{1}{z}=6</math>.
But the Cauchy-Swarsz inequality gives
<math>(x+y+z)(\frac{9}{x}+\frac{4}{y}+\frac{1}{z})</math>
<math>\geq(\sqrt{x\times\frac{9}{x}}+\sqrt{y\times\frac{4}{y}}
+\sqrt{z\times\frac{1}{z}})^2</math>
<math>=(3+2+1)^2=36.</math>
Since <math>(x+y+z)(\frac{9}{x}+\frac{4}{y}+\frac{1}{z})=6\times6=36</math>, we have the equality case, i.e. <math>x:y:z=\frac{9}{x}:\frac{4}{y}:\frac{1}{z}</math>, or <math>x:y:z=3:2:1</math>, or <math>(x,y,z)=(3,2,1)</math>, and <math>xyz=\boxed{6}</math>.
~AbbyWong