Tricky problem(Old book ex:1.75): (BOMC)
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Given: $a,b,c > 0$ ; $a+b+c=3$ ; $abc=1$
Prove that,$\frac{ab}{(a^2+b)(a+b^2)}+\frac{bc}{(b^2+c)(b+c^2)}+\frac{ca}{(c^2+a)(c+a^2)}\leq\frac{3}{4}$
Prove that,$\frac{ab}{(a^2+b)(a+b^2)}+\frac{bc}{(b^2+c)(b+c^2)}+\frac{ca}{(c^2+a)(c+a^2)}\leq\frac{3}{4}$
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......(-$from$ "$THE$ $UGLY$ $TRUTH$" )
- zadid xcalibured
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Re: Tricky problem(Old book ex:1.75): (BOMC)
this is the equality case of AMGM so a=b=c=1 and we r done.
Re: Tricky problem(Old book ex:1.75): (BOMC)
Solution:
Last edited by *Mahi* on Fri Oct 28, 2011 7:43 pm, edited 1 time in total.
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Re: Tricky problem(Old book ex:1.75): (BOMC)
I'd love to see the trick though...
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Re: Tricky problem(Old book ex:1.75): (BOMC)
Zadid has given the trick... (Sorry as i have approved the post so lately.) Welcome Zadid...
Trick was in given conditions. For the given condition $a=b=c=1$. Direct equality....
Trick was in given conditions. For the given condition $a=b=c=1$. Direct equality....
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......(-$from$ "$THE$ $UGLY$ $TRUTH$" )
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Re: Tricky problem(Old book ex:1.75): (BOMC)
Factorizing silly mistakes....*Mahi* wrote:Solution:
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......(-$from$ "$THE$ $UGLY$ $TRUTH$" )
Re: Tricky problem(Old book ex:1.75): (BOMC)
The proof still holds in that way... it's about the degree...not what the form is...
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- FahimFerdous
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Re: Tricky problem(Old book ex:1.75): (BOMC)
Ha ha ha! Really a nice one!
Your hot head might dominate your good heart!
Re: Tricky problem(Old book ex:1.75): (BOMC)
And whatever, YOU should have seen that the factorization mistake doesn't matter at all, the part of taking the GM is all about the degree of the polynomial.
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- bristy1588
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Re: Tricky problem(Old book ex:1.75): (BOMC)
Problem 1.69 of Old book:
Can anyone just show me the steps? How do we apply Rearrangement here?
Can anyone just show me the steps? How do we apply Rearrangement here?
Bristy Sikder