INFINITY SOLUTIONS
-
MATHPRITOM
- Posts:190
- Joined:Sat Apr 23, 2011 8:55 am
- Location:Khulna
Prove that, the equation $ x^2+y^2z+z^2x=3xyz $ has infinity positive solutions.
-
sakibtanvir
- Posts:188
- Joined:Mon Jan 09, 2012 6:52 pm
- Location:24.4333°N 90.7833°E
Re: INFINITY SOLUTIONS
\[x^2+y^2z+z^2x>3xyz\]
The ineq. becomes eq. when \[x^{2}=y^{2}z=z^{2}x\]
And ;it follows the condition if,\[\sqrt{x}=z,(\sqrt{x})^{3}=y^2\]
So,Any $x$ for which,\[\sqrt[6]{x}=k\]
Where $k$ is an integer can breed the equation.Hence it is proved.
The ineq. becomes eq. when \[x^{2}=y^{2}z=z^{2}x\]
And ;it follows the condition if,\[\sqrt{x}=z,(\sqrt{x})^{3}=y^2\]
So,Any $x$ for which,\[\sqrt[6]{x}=k\]
Where $k$ is an integer can breed the equation.Hence it is proved.
An amount of certain opposition is a great help to a man.Kites rise against,not with,the wind.