Hint:
PEN-2007-5
- Tahmid Hasan
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- Location:Khulna,Bangladesh.
Let $x,y$ be positive integers such that $xy \mid x^2+y^2+1$.Show that $\frac {x^2+y^2+1}{xy}=3$.
Hint:
Hint:
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- Tahmid Hasan
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Re: PEN-2007-5
here's another solution without flipping roots whatsoever but using only descent and inequality.
Last edited by Tahmid Hasan on Mon Apr 02, 2012 6:57 pm, edited 1 time in total.
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- Tahmid Hasan
- Posts:665
- Joined:Thu Dec 09, 2010 5:34 pm
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Re: PEN-2007-5
well these problems can also be solved using the same method i've shown without flipping roots.you should try it too .And i learned this technique from "An Introduction to Diophantine Equations" by Titu Andreescu and Dorin Andrica.zadid xcalibured wrote:
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- FahimFerdous
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Re: PEN-2007-5
@Tahmid: your proof is called Vieta Jumping if you don't know then.
Your hot head might dominate your good heart!
- zadid xcalibured
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Re: PEN-2007-5
he knows actually.
- Tahmid Hasan
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Re: PEN-2007-5
well Fahim vaiya,as far as i know if the equation $x^2+ax+b=0$ has real roots $p,q$ then by vieta's formula $pq=b,-a=p+q$ and the principle of vieta jumping is based on this and descent.And vieta jumping is also called root flipping accordong to wiki, and the phrase 'root flipping' sounds more 'cruxy' to me
Last edited by Tahmid Hasan on Mon Apr 02, 2012 7:28 pm, edited 1 time in total.
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- zadid xcalibured
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Re: PEN-2007-5
root flipping is just one kind of descent.@tahmid just using vieta formula.