## National2009/4

For students of class 6-8 (age 12 to 14)
sakibtanvir
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### National2009/4

$x^2-8xy+9y^2-16y+10$Find the least possible value of the expression.$(x,y)\in R$
An amount of certain opposition is a great help to a man.Kites rise against,not with,the wind.

sakibtanvir
Posts: 188
Joined: Mon Jan 09, 2012 6:52 pm
Location: 24.4333°N 90.7833°E

### 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
Posts: 665
Joined: Thu Dec 09, 2010 5:34 pm

### Re: National2009/4

what is the least value of a square number?
(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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sakibtanvir
Posts: 188
Joined: Mon Jan 09, 2012 6:52 pm
Location: 24.4333°N 90.7833°E

### 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
Posts: 665
Joined: Thu Dec 09, 2010 5:34 pm

### Re: National2009/4

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

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.
$\sum_{k=0}^{n-1}e^{\frac{2 \pi i k}{n}}=0$
Using $L^AT_EX$ and following the rules of the forum are very easy but really important, too.Please co-operate.

sakibtanvir
Posts: 188
Joined: Mon Jan 09, 2012 6:52 pm
Location: 24.4333°N 90.7833°E

### Re: National2009/4

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

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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When you go down, when you go down down......
(-$from$ "$THE$ $UGLY$ $TRUTH$" )

samiul_samin
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### Re: National2009/4

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!