Determine all ordered pair of positive integers $(x,y)$ so that
$\frac{x^2+2y^2}{2x^2+y^2}$ is the square of an integer.
Perfect Square ratio
 Fm Jakaria
 Posts: 79
 Joined: Thu Feb 28, 2013 11:49 pm
Perfect Square ratio
You cannot say if I fail to recite
the umpteenth digit of PI,
Whether I'll live  or
whether I may, drown in tub and die.
the umpteenth digit of PI,
Whether I'll live  or
whether I may, drown in tub and die.
Re: Perfect Square ratio
If $x\neq y$ then \[\mathbb{N}\ni\dfrac{x^2+2y^2}{2x^2+y^2}=1+\dfrac{y^2x^2}{2x^2+y^2}~\Rightarrow ~ 2x^2+y^2\mid y^2x^2\] \[\Rightarrow ~ 2x^2+y^2\le y^2x^2 ~\Rightarrow ~ 3x^2\le 0 ~\Rightarrow ~ x=0\not\in\mathbb{N} \] hence we must have $x=y$, thus all pairs $\mathbb{N}^2\ni (x,y)=(n,n)$ works.
 What is the value of the contour integral around Western Europe?
 Zero.
 Why?
 Because all the poles are in Eastern Europe.
Revive the IMO marathon.
 Zero.
 Why?
 Because all the poles are in Eastern Europe.
Revive the IMO marathon.

 Posts: 107
 Joined: Sun Dec 12, 2010 10:46 am
Re: Perfect Square ratio
The reasoning falls apart if $ x $ is greater than $ y $.
Re: Perfect Square ratio
which is impossible.mutasimmim wrote:The reasoning falls apart if $ x $ is greater than $ y $
 What is the value of the contour integral around Western Europe?
 Zero.
 Why?
 Because all the poles are in Eastern Europe.
Revive the IMO marathon.
 Zero.
 Why?
 Because all the poles are in Eastern Europe.
Revive the IMO marathon.
Re: Perfect Square ratio
And that can be shown in two different ways.
 What is the value of the contour integral around Western Europe?
 Zero.
 Why?
 Because all the poles are in Eastern Europe.
Revive the IMO marathon.
 Zero.
 Why?
 Because all the poles are in Eastern Europe.
Revive the IMO marathon.
Re: Perfect Square ratio
To avoid the confusion, we may write that $x^{2}+2y^{2}\leq y^{2}x^{2}$. Square it, simplify it, then we can get $3x^{2}(x^{2}+2y^{2})\leq 0$.
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