Exercise-1.15(new book) (BOMC-2011)

Discussion on Bangladesh National Math Camp
sm.joty
Posts:327
Joined:Thu Aug 18, 2011 12:42 am
Location:Dhaka
Exercise-1.15(new book) (BOMC-2011)
Let a,b,c be positive real numbers such that abc=1. Prove that
$\frac{ab}{a^{5}+b^{5}+ab}+\frac{bc}{b^{5}+c^{5}+bc}+\frac{ca}{c^{5}+a^{5}+ca} \leq1$

হার জিত চিরদিন থাকবেই
তবুও এগিয়ে যেতে হবে.........
বাধা-বিঘ্ন না পেরিয়ে
বড় হয়েছে কে কবে.........

sourav das
Posts:461
Joined:Wed Dec 15, 2010 10:05 am
Location:Dhaka
Contact:

Re: Exercise-1.15(new book) (BOMC-2011)

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:
You spin my head right round right round,
When you go down, when you go down down......
(-$from$ "$THE$ $UGLY$ $TRUTH$" )

FahimFerdous
Posts:176
Joined:Thu Dec 09, 2010 12:50 am

Re: Exercise-1.15(new book) (BOMC-2011)

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.

sourav das
Posts:461
Joined:Wed Dec 15, 2010 10:05 am
Location:Dhaka
Contact:

Re: Exercise-1.15(new book) (BOMC-2011)

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......
(-$from$ "$THE$ $UGLY$ $TRUTH$" )

nafistiham
Posts:829
Joined:Mon Oct 17, 2011 3:56 pm
Location:24.758613,90.400161
Contact:

Re: Exercise-1.15(new book) (BOMC-2011)

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.
$\sum_{k=0}^{n-1}e^{\frac{2 \pi i k}{n}}=0$
Using $L^AT_EX$ and following the rules of the forum are very easy but really important, too.Please co-operate.

*Mahi*
Posts:1175
Joined:Wed Dec 29, 2010 12:46 pm
Location:23.786228,90.354974
Contact:

Re: Exercise-1.15(new book) (BOMC-2011)

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.

Use $L^AT_EX$, It makes our work a lot easier!

Ashfaq Uday
Posts:21
Joined:Tue Sep 27, 2011 12:18 am

Re: Exercise-1.15(new book) (BOMC-2011)

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???

Ashfaq Uday
Posts:21
Joined:Tue Sep 27, 2011 12:18 am

Re: Exercise-1.15(new book) (BOMC-2011)

oww. so silly of me. abc=1 condition was missed by me. Now i understand sourov's approach. Bt is my process right??

*Mahi*
Posts:1175
Joined:Wed Dec 29, 2010 12:46 pm
Location:23.786228,90.354974
Contact:

Re: Exercise-1.15(new book) (BOMC-2011)

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???
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.
Use $L^AT_EX$, It makes our work a lot easier!