National2009/4

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sakibtanvir
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National2009/4

Unread post by sakibtanvir » Sun Jan 15, 2012 9:43 pm

\[x^2-8xy+9y^2-16y+10\]Find the least possible value of the expression.\[(x,y)\in R\]
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sakibtanvir
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Re: National2009/4

Unread post by sakibtanvir » Mon Jan 16, 2012 11:31 am

Is there anyone to help? :?
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Tahmid Hasan
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Re: National2009/4

Unread post by Tahmid Hasan » Mon Jan 16, 2012 11:49 am

what is the least value of a square number? ;)
(thanks @Tiham,that was a silly mistake :oops: )
Last edited by Tahmid Hasan on Tue Jan 17, 2012 7:51 pm, edited 1 time in total.
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sakibtanvir
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Re: National2009/4

Unread post by sakibtanvir » Mon Jan 16, 2012 2:04 pm

let it $P(x,y)$.The least possible value of $(x^2,y^2)$ would be $(1,0)$ or $(0,1)$.So the possible answer will be $P(1,0)$ or $P(0,1)$.Now we get two values of the polynomial.$3$ and $11$ so the answer is $3$.Is my solution correct?? :cry:I am so confused :? :? :roll:
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Tahmid Hasan
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Re: National2009/4

Unread post by Tahmid Hasan » Mon Jan 16, 2012 4:21 pm

well,i faced this problem in BdMO.tried to solve it in trial and error but failed.
Again use the hint :P
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nafistiham
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Re: National2009/4

Unread post by nafistiham » Tue Jan 17, 2012 7:03 pm

Tahmid Hasan wrote:what is the highest value of a square number? ;)
Tahmid, didn't you want to say 'least' ?

try to express the expression as a summation of some squares and a constant. ;)
\[\sum_{k=0}^{n-1}e^{\frac{2 \pi i k}{n}}=0\]
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sakibtanvir
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Re: National2009/4

Unread post by sakibtanvir » Sun Jan 29, 2012 7:06 pm

An amount of certain opposition is a great help to a man.Kites rise against,not with,the wind.

sourav das
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Re: National2009/4

Unread post by sourav das » Sun Jan 29, 2012 9:40 pm

2009 was the first national for me. I don't remember clearly but i think i managed to find a term that shows that the expression can't have any smallest value. But in the hall i thought "What have i done...." I was sad and give up. And after seeing the solution in Newspaper...........
Ok, the expression can be written in form of: $(x-4y)^2-(\sqrt {7} y+\frac{8}{\sqrt{7}})^2+\frac{64}{7}+10$
Let's make it a little more interesting: viewtopic.php?f=21&t=1629&p=8379#p8379
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samiul_samin
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Re: National2009/4

Unread post by samiul_samin » Sun Feb 24, 2019 12:34 am

sakibtanvir wrote:
Sun Jan 15, 2012 9:43 pm
\[x^2-8xy+9y^2-16y+10\]Find the least possible value of the expression.\[(x,y)\in R\]
This is BdMO National Junior $2009$ Problem no $4$.Very tough one!

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