lowest Value

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samiul_samin
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lowest Value

Unread post by samiul_samin » Wed Feb 14, 2018 12:19 am

$a=p/q$
$b=q/r$
$c=r/p$
p,q,r are Positive Integer.
Find the lowest value of
#$(4+a)(4+b)(4+c)$
And
#$(1+a)(1+b)(1+c)$
Source: bigganchinta december 2016

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Phlembac Adib Hasan
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Re: lowest Value

Unread post by Phlembac Adib Hasan » Wed Feb 14, 2018 4:59 am

Hint:
1.
$abc=1$. Now apply AM-GM.
2.
#2 directly follows from AM-GM. For #1, can you find a clever way to apply AM-GM?
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samiul_samin
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Re: lowest Value

Unread post by samiul_samin » Wed Feb 14, 2018 3:35 pm

Phlembac Adib Hasan wrote:
Wed Feb 14, 2018 4:59 am
Hint:
1.
$abc=1$. Now apply AM-GM.
2.
#2 directly follows from AM-GM. For #1, can you find a clever way to apply AM-GM?

THANKS ADIB VAIA.
Answer
1.
$a+b+c\geq 3$ so,the lowest value is$ \fbox {125}$
2.
the lowest value is $\fbox 8$
Is it a clever way to use $a+1/a \geq2$ to get the lowest value of# 1?

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