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- Feb 14, 2012

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Given $a,b,c,d$ are real numbers such that

$ab+cd=4$

$ac+bd=8$

Find the maximum value of $abcd$.

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Remember to read the POTW submission guidelines to find out how to submit your answers!

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- Thread starter
- Admin
- #1

- Feb 14, 2012

- 3,894

-----

Given $a,b,c,d$ are real numbers such that

$ab+cd=4$

$ac+bd=8$

Find the maximum value of $abcd$.

-----

Remember to read the POTW submission guidelines to find out how to submit your answers!

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- #2

- Feb 14, 2012

- 3,894

1. castor28

2. Olinguito

3. kaliprasad

Solution from castor28 :

$$

X^2 - (ab+cd)X + (ab)(cd) = 0

$$

Since this equation has real roots and $ab+cd=4$, we must have:

$$

4(abcd) \le (ab+cd)^2 = 16

$$

Which implies $abcd\le 4$.

Using a similar argument, the relation $ac+bd=8$ gives the weaker (larger) bound $abcd\le16$.

It remains to show that the maximum ($4$) can be attained. We let $a=1, b=2$ and require $cd=2$. This gives the system of equations:

\begin{align*}

c+2d&=8\\

cd&=2

\end{align*}

with real solutions $c=4\pm2\sqrt3,d=2\mp\sqrt3$.

The maximum value of $abcd$ is therefore $\mathbf{4}$.

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