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

Discussion on Bangladesh National Math Camp
User avatar
sm.joty
Posts:327
Joined:Thu Aug 18, 2011 12:42 am
Location:Dhaka
Exercise-1.15(new book) (BOMC-2011)

Unread post by sm.joty » Thu Oct 27, 2011 2:04 pm

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.
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)

Unread post by sourav das » Thu Oct 27, 2011 3:41 pm

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:
$a^5+b^5=(a+b)(a^4-a^3b+a^2b^2-ab^3+b^4)$; $a^4+b^4 \geq a^3b +b^3a$; So $a^5 + b^5 + ab \geq ab(ab(a+b)+1)$
You spin my head right round right round,
When you go down, when you go down down......
(-$from$ "$THE$ $UGLY$ $TRUTH$" )

User avatar
FahimFerdous
Posts:176
Joined:Thu Dec 09, 2010 12:50 am
Location:Mymensingh, Bangladesh

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

Unread post by FahimFerdous » Thu Oct 27, 2011 8:10 pm

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.
Your hot head might dominate your good heart!

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

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

Unread post by sourav das » Thu Oct 27, 2011 9:22 pm

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

User avatar
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)

Unread post by nafistiham » Thu Oct 27, 2011 9:30 pm

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.
Introduction:
Nafis Tiham
CSE Dept. SUST -HSC 14'
http://www.facebook.com/nafistiham
nafistiham@gmail

User avatar
*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)

Unread post by *Mahi* » Thu Oct 27, 2011 9:33 pm

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.
Please read Forum Guide and Rules before you post.

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

Nur Muhammad Shafiullah | Mahi

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

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

Unread post by Ashfaq Uday » Sun Nov 06, 2011 9:32 pm

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)

Unread post by Ashfaq Uday » Mon Nov 07, 2011 6:35 am

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

User avatar
*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)

Unread post by *Mahi* » Mon Nov 07, 2011 7:25 am

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.
Please read Forum Guide and Rules before you post.

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

Nur Muhammad Shafiullah | Mahi

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

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

Unread post by Ashfaq Uday » Mon Nov 07, 2011 1:03 pm

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??

Post Reply