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Nirjhor
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Minimize!

Unread post by Nirjhor » Wed Mar 04, 2015 12:28 am

Let $a,b,c>0$ satisfy $\left(a+b+c\right)^2+\left(a^2+b^2+c^2\right)\le 4$. Find the minimum value of \
Last edited by Nirjhor on Thu Mar 05, 2015 12:41 am, edited 2 times in total.
- What is the value of the contour integral around Western Europe?

- Zero.

- Why?

- Because all the poles are in Eastern Europe.


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Tahmid
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Joined:Wed Mar 20, 2013 10:50 pm

Re: Minimize!

Unread post by Tahmid » Wed Mar 04, 2015 12:37 am

Nirjhor wrote: $\displaystyle\sum_{\text{cyc}} \dfrac{ab+1}{(a+b)^2}$.
$cyc$ means? :|

Nirjhor
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Re: Minimize!

Unread post by Nirjhor » Wed Mar 04, 2015 12:48 am

It means summing all versions of the inner expression with variables cyclically permuted. I've edited the statement for clarity.
- 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.

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Phlembac Adib Hasan
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Re: Minimize!

Unread post by Phlembac Adib Hasan » Thu Mar 05, 2015 7:20 pm

Let $a+b=x,b+c=y,c+a=z$. The given condition becomes $4\ge x^2+y^2+z^2\quad (1)$
\[\begin{align*}S &= \sum_{cyc} \frac{(x-y+z)(y-z+x)+4}{4x^2}\\
& = \frac 3 4 + \sum_{cyc} \frac{4-(y-z)^2}{4x^2}\\
& \ge \frac 3 4 +\sum_{cyc} \frac{x^2+2yz}{4x^2}\quad [\text{Using (1)}]\\
& \ge \frac 3 4 + \frac 3 4 + \frac 2 4 \cdot 3\sqrt[3]{\frac{yz}{x^2}\cdot \frac{zx}{y^2}\cdot \frac{xy}{z^2}}\\
& =3\end{align*}\]
So the minimum is $3$ which is achieved by $a=b=c=\frac 1 {\sqrt 3}$
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Nirjhor
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Re: Minimize!

Unread post by Nirjhor » Thu Mar 05, 2015 10:39 pm

My solution.
\[\begin{eqnarray}
4S=\sum_{\text{cyc}}\dfrac{4ab+4}{(a+b)^2}&\ge&\sum_{\text{cyc}}\dfrac{4ab+(a+b+c)^2+(a^2+b^2+c^2)}{(a+b)^2} \\
&=& \sum_{\text{cyc}}\dfrac{2(a+b)^2+2(ab+bc+ca+c^2)}{(a+b)^2} \\
&=& 6+2\sum_{\text{cyc}}\dfrac{(b+c)(c+a)}{(a+b)^2} \\
&\ge & 6+6\sqrt[3]{\prod_{\text{cyc}}\dfrac{(b+c)(c+a)}{(a+b)^2}} =12.\\
\end{eqnarray}\] So $S\ge 3$ with equality at $a=b=c=1/\sqrt 3$.
- 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.

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