Exercise1.15(new book) (BOMC2011)
Let a,b,c be positive real numbers such that abc=1. Prove that
\[\frac{ab}{a^{5}+b^{5}+ab}+\frac{bc}{b^{5}+c^{5}+bc}+\frac{ca}{c^{5}+a^{5}+ca} \leq1\]
Please give some hints.
\[\frac{ab}{a^{5}+b^{5}+ab}+\frac{bc}{b^{5}+c^{5}+bc}+\frac{ca}{c^{5}+a^{5}+ca} \leq1\]
Please give some hints.
হার জিত চিরদিন থাকবেই
তবুও এগিয়ে যেতে হবে.........
বাধাবিঘ্ন না পেরিয়ে
বড় হয়েছে কে কবে.........
তবুও এগিয়ে যেতে হবে.........
বাধাবিঘ্ন না পেরিয়ে
বড় হয়েছে কে কবে.........

 Posts:461
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Re: Exercise1.15(new book) (BOMC2011)
I prefer not to ask for hints until you have tried to solve it at least 4.5 hours. You are solving problems, not exercises. I know that it is always very tempting to see the tricks behind any problem. But it feels much better if you find it out all by yourself.
Those who have tried at least 4.5 hours:
Those who have tried at least 4.5 hours:
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......($from$ "$THE$ $UGLY$ $TRUTH$" )
 FahimFerdous
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Re: Exercise1.15(new book) (BOMC2011)
Sourav, 4.5 hours is too much now because there are other problems too.
But people, try at least for 2.5 to 3 hours. Otherwise, you won't get the patience that's needed to solve a problem.
But people, try at least for 2.5 to 3 hours. Otherwise, you won't get the patience that's needed to solve a problem.
Your hot head might dominate your good heart!

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Re: Exercise1.15(new book) (BOMC2011)
Agree with Fahim. But when you'll have sufficient time, don't give up easily.
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......($from$ "$THE$ $UGLY$ $TRUTH$" )
 nafistiham
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Re: Exercise1.15(new book) (BOMC2011)
i think everyone should decide how much time they should work on a problem.everyone has due problems giving this time to problem solving.but,the more one works on a problem, the better.
\[\sum_{k=0}^{n1}e^{\frac{2 \pi i k}{n}}=0\]
Using $L^AT_EX$ and following the rules of the forum are very easy but really important, too.Please cooperate.
Using $L^AT_EX$ and following the rules of the forum are very easy but really important, too.Please cooperate.
Re: Exercise1.15(new book) (BOMC2011)
I agree with Fahim bhai. This is a very recent time IMO problem and for someone who is new to inequality, if the solution doesn't come within 2.5 hrs, giving any more time is waste right now (of course they can try it later.)
BTW, it would be better if everyone posts their approach here. Then others can propose improvisations on the unfinished solution.
BTW, it would be better if everyone posts their approach here. Then others can propose improvisations on the unfinished solution.
Please read Forum Guide and Rules before you post.
Use $L^AT_EX$, It makes our work a lot easier!
Nur Muhammad Shafiullah  Mahi
Use $L^AT_EX$, It makes our work a lot easier!
Nur Muhammad Shafiullah  Mahi

 Posts:21
 Joined:Tue Sep 27, 2011 12:18 am
Re: Exercise1.15(new book) (BOMC2011)
I wanted to prove each of the fractions be less than or equal to \[1/3\]
WLOG let \[a\geq b\Rightarrow a/b\geq 1\]
so,\[ab/\left ( b^5\left ( 1+\left ( a^5/b^5 \right ) \right ) +ab\right )\leq ab/\left ( 2b^5+ab \right )=a/\left ( a+2b^4 \right )\]
which is \[\leq 1/3\] for any positive integer.I cant prove it for real number as setting \[a=.2,b=.1,c=.3\] (just an example) gives an absurd result. I must be missing something. Can somebody help me finish this? and Is my process right???
WLOG let \[a\geq b\Rightarrow a/b\geq 1\]
so,\[ab/\left ( b^5\left ( 1+\left ( a^5/b^5 \right ) \right ) +ab\right )\leq ab/\left ( 2b^5+ab \right )=a/\left ( a+2b^4 \right )\]
which is \[\leq 1/3\] for any positive integer.I cant prove it for real number as setting \[a=.2,b=.1,c=.3\] (just an example) gives an absurd result. I must be missing something. Can somebody help me finish this? and Is my process right???

 Posts:21
 Joined:Tue Sep 27, 2011 12:18 am
Re: Exercise1.15(new book) (BOMC2011)
oww. so silly of me. abc=1 condition was missed by me. Now i understand sourov's approach. Bt is my process right??
Re: Exercise1.15(new book) (BOMC2011)
See closely, your proof requires $a\geq b$, $b \geq c$ and $c \geq a$, but all three of them can't be true at the same time.Ashfaq Uday wrote:I wanted to prove each of the fractions be less than or equal to \[1/3\]
WLOG let \[a\geq b\Rightarrow a/b\geq 1\]
so,\[ab/\left ( b^5\left ( 1+\left ( a^5/b^5 \right ) \right ) +ab\right )\leq ab/\left ( 2b^5+ab \right )=a/\left ( a+2b^4 \right )\]
which is \[\leq 1/3\] for any positive integer.I cant prove it for real number as setting \[a=.2,b=.1,c=.3\] (just an example) gives an absurd result. I must be missing something. Can somebody help me finish this? and Is my process right???
Please read Forum Guide and Rules before you post.
Use $L^AT_EX$, It makes our work a lot easier!
Nur Muhammad Shafiullah  Mahi
Use $L^AT_EX$, It makes our work a lot easier!
Nur Muhammad Shafiullah  Mahi

 Posts:21
 Joined:Tue Sep 27, 2011 12:18 am
Re: Exercise1.15(new book) (BOMC2011)
could somebody please explain me why a discriminant has to be negative for a quadratic function to be positive?? when a quadratic function is positive, does it refer that it's value is positive or the signs are all positive??