$\textbf{Calculate real values of $\mathbf{x}$ in\\\\ $\mathbf{27^x+125^x+343^x+1331^x=105^x+165^x+231^x+385^x}$}
21the equation is
(3)3x + (5)3x + (7)3x + (11)3x = (3*5*7)x + (3*11*5)x + (3*7*11)x + ( 5*7*11)x
Let 3x= a,5x=b , 7x=c, 11x= d (a,b,c,d all are obviously positive)
then the equation becomes
a3+ b3 + c3 + d3 = abc + bcd + cda + dab
But by muirhead inequatlity (or by AM-GM or Re-arrangement)
a3+ b3 + c3 + d3≥ abc + bcd + cda + dab
with equality iff a=b=c=d
which means 3x= 5x= 7x=11x
So x = 0 is the only solution ..
62great proof.. couldnt move beyond the 4th step...
somehow rearrangement never came to my mind..
1u know this is the problem why people keep complaiaing about higher level questions....most of people(including me)would have started solving the problem and reached till here
the equation is
(3)3x + (5)3x + (7)3x + (11)3x = (3*5*7)x + (3*11*5)x + (3*7*11)x + ( 5*7*11)x
Let 3x= a,5x=b , 7x=c, 11x= d (a,b,c,d all are obviously positive)
then the equation becomes
a3+ b3 + c3 + d3 = abc + bcd + cda + dab
and got BLOCKED
now for further learning outcomes we need to know what kind of working is this But by muirhead inequatlity (or by AM-GM or Re-arrangement).....this is the LIFE of the problem....maybe trivial for the one posting the solution...but for those who want to learn new things u need to explain what is But by "muirhead inequatlity (or by AM-GM or Re-arrangement) in ur post"...else it would serve no LEARNING PURPOSE for others!
62Chinmaya.. the point that you make is very good.. and i absolutely agree to it...
This has been one very imprortant reason why a lot of users have not been able to like this site.. and this is also one reason why many others have loved it...
The thing I think is that those who are "interested" will always try to search or ask about rearrangement and then they will get the answers.... those who dont are not interested enough to find out why....
The most important thing for an online forum is for people to ask politely and hope to get answers....
1hmm...i agree this way atleast we know "what to search for"...(m doing that right now)...hopefully we would get good search results for named inequality but for rearrangement type search we are going to get bewildring results until we don't know what kind of rearrangement is being referred to...and if we already knew there seems no point in asking,isn't it?
62an interesting post which kind of proves what i am saying ;)
http://www.targetiit.com/iit-jee-forum/posts/show-that-18170.html
This is the same person who posted this solution, who was asking for the same inequality :D
62http://www.artofproblemsolving.com/Wiki/index.php/Category:Inequality
read this... good source for inequalities...
1Ok ok ok...got it now that rearrangement is a type of inequality
I was wondering wat is rearrangement for last 20 minutes[2]
1708I am also struk at that point.
Now thanks Shubhodip for Great solution.
also thanks for encouraging me to posting Questions.
Also thanks Nishant Sir.
1a few interesting points from site whih nishant bhaiya told
How to Write a Solution
Readers Are Not Interpreters
by Richard Rusczyk
The first thing a reader sees on your paper isn't the structure of your solution. It isn't the answer, it isn't the words you choose. It's how the solution sits on the paper. If the reader has to decipher scrawl, you're going to lose him.
http://www.artofproblemsolving.com/Resources/articles.php?page=htw.readers
How to Write a Solution
Solution Readers, not Mindreaders
by Richard Rusczyk
A full solution does not just mean a correct answer. You should justify every notable step of your solution. An experienced reader should never wonder 'Why is that true?' while reading your solution. She should also never be left in doubt as to whether or not you know why it is true.
http://www.artofproblemsolving.com/Resources/articles.php?page=htw.mindreaders