## Perfect Square ratio

For discussing Olympiad Level Number Theory problems
Fm Jakaria
Posts: 79
Joined: Thu Feb 28, 2013 11:49 pm

### Perfect Square ratio

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.
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.

Nirjhor
Posts: 136
Joined: Thu Aug 29, 2013 11:21 pm
Location: Varies.

### Re: Perfect Square ratio

If $x\neq y$ then $\mathbb{N}\ni\dfrac{x^2+2y^2}{2x^2+y^2}=1+\dfrac{y^2-x^2}{2x^2+y^2}~\Rightarrow ~ 2x^2+y^2\mid y^2-x^2$ $\Rightarrow ~ 2x^2+y^2\le y^2-x^2 ~\Rightarrow ~ 3x^2\le 0 ~\Rightarrow ~ x=0\not\in\mathbb{N}$ 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.

mutasimmim
Posts: 107
Joined: Sun Dec 12, 2010 10:46 am

### Re: Perfect Square ratio

The reasoning falls apart if $x$ is greater than $y$.

Nirjhor
Posts: 136
Joined: Thu Aug 29, 2013 11:21 pm
Location: Varies.

### Re: Perfect Square ratio

mutasimmim wrote:The reasoning falls apart if $x$ is greater than $y$
which is impossible.
- 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.

Nirjhor
Posts: 136
Joined: Thu Aug 29, 2013 11:21 pm
Location: Varies.

### 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.

SANZEED
Posts: 550
Joined: Wed Dec 28, 2011 6:45 pm
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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