Perfect Square ratio

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

Perfect Square ratio

Unread post by Fm Jakaria » Sun Nov 09, 2014 4:06 pm

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

Unread post by Nirjhor » Mon Nov 10, 2014 3:32 am

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

Unread post by mutasimmim » Mon Nov 10, 2014 7:06 pm

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

Unread post by Nirjhor » Mon Nov 10, 2014 7:46 pm

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

Unread post by Nirjhor » Mon Nov 10, 2014 7:50 pm

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.

User avatar
SANZEED
Posts: 550
Joined: Wed Dec 28, 2011 6:45 pm
Location: Mymensingh, Bangladesh

Re: Perfect Square ratio

Unread post by SANZEED » Thu Nov 13, 2014 2:38 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 $3x^{2}(x^{2}+2y^{2})\leq 0$.
$\color{blue}{\textit{To}} \color{red}{\textit{ problems }} \color{blue}{\textit{I am encountering with-}} \color{green}{\textit{AVADA KEDAVRA!}}$

Post Reply