National2009/4

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National2009/4
\[x^28xy+9y^216y+10\]Find the least possible value of the expression.\[(x,y)\in R\]
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Re: National2009/4
Is there anyone to help?
An amount of certain opposition is a great help to a man.Kites rise against,not with,the wind.
 Tahmid Hasan
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Re: National2009/4
what is the least value of a square number?
(thanks @Tiham,that was a silly mistake )
(thanks @Tiham,that was a silly mistake )
Last edited by Tahmid Hasan on Tue Jan 17, 2012 7:51 pm, edited 1 time in total.
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Re: National2009/4
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?? I am so confused
An amount of certain opposition is a great help to a man.Kites rise against,not with,the wind.
 Tahmid Hasan
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Re: National2009/4
well,i faced this problem in BdMO.tried to solve it in trial and error but failed.
Again use the hint
Again use the hint
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 nafistiham
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Re: National2009/4
Tahmid, didn't you want to say 'least' ?Tahmid Hasan wrote:what is the highest value of a square number?
try to express the expression as a summation of some squares and a constant.
\[\sum_{k=0}^{n1}e^{\frac{2 \pi i k}{n}}=0\]
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Using $L^AT_EX$ and following the rules of the forum are very easy but really important, too.Please cooperate.

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Re: National2009/4
http://www.wolframalpha.com/input/?i=x% ... 16*y%2B10
Hyperbolic Paraboloid.
Hyperbolic Paraboloid.
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Re: National2009/4
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: $(x4y)^2(\sqrt {7} y+\frac{8}{\sqrt{7}})^2+\frac{64}{7}+10$
Let's make it a little more interesting: viewtopic.php?f=21&t=1629&p=8379#p8379
Ok, the expression can be written in form of: $(x4y)^2(\sqrt {7} y+\frac{8}{\sqrt{7}})^2+\frac{64}{7}+10$
Let's make it a little more interesting: viewtopic.php?f=21&t=1629&p=8379#p8379
You spin my head right round right round,
When you go down, when you go down down......($from$ "$THE$ $UGLY$ $TRUTH$" )
When you go down, when you go down down......($from$ "$THE$ $UGLY$ $TRUTH$" )
 samiul_samin
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Re: National2009/4
This is BdMO National Junior $2009$ Problem no $4$.Very tough one!sakibtanvir wrote: ↑Sun Jan 15, 2012 9:43 pm\[x^28xy+9y^216y+10\]Find the least possible value of the expression.\[(x,y)\in R\]