go ahead and answer its simple
Find the sum of all integral solutions of
x 3 + 5 y 3 + 2 5 z 3 − 1 5 x y z = 0
x , y , z are positive integers
Please post your methods also
This question is flagged because there are several issues with it.
The answer is 0.
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1 solution
good method!
beautifully done
Perhaps you should have mentioned positive integers to apply AM-GM method.
Can you explain why "difference of all integral solutions" mean?
Also, can you explain why "Given AM=GM, therefore x=y=z=0"? How does that follow (note: you are missing a step).
In the 2nd method, it is best that you do not abuse notation, and instead choose difference variables. Note that once again, the solution is not "x=y=z=0", but "x=y=z". Why must it be equal to 0?
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A.M = G.M
this is only possible when all the numbers are equal
therefore x = 5 y 3 = 2 5 z 3
this is only possible when x=y=z=0
here we are getting only one integral solution therefore
x - y - z or y - x - z or z - x - y are equal,so whom so ever we subtract for the other the answer remains constant
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You should still give a short explanation of why "this is only possible when x=y=z=0". For example, if we were looking at real number solutions, then there are many other solutions like x = 5 , y = 3 2 5 , z = 3 5 . What is so special about the integers that makes there only be 1 solution?
Note that your explanation in the last line is not true.
I still do not understand what "find the difference of all integer solutions" means.
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@Calvin Lin – sir I thought that since we are getting this x = 5 y 3 = 2 5 z 3
therefore the integers satisfying it is only 0
and since we are getting only 0 as the solution therefor difference of any term will be 0
please I am not able to understand what is wrong , can you please elaborate
I am changing the difference to the product
thank you
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@U Z – Here are the issues that I have:
1) "AM = GM" only implies x 3 = 5 y 3 = 2 5 z 3 and not x = y = z = 0 .
2) You need to use the fact that 3 5 is irrational and x 3 = 5 y 3 = 2 5 z 3 where all are integers, to conclude that x = y = z = 0 .
3) In the previous phrasing, find the difference of all integer solutions is unclear, since we do not know what "the difference of the solution (1, 2, 3)" would be. Just because the answer happens to be (0,0,0) doesn't make your phrasing "immediately clear".
4) Now, there are no solutions to your question. The empty product is defined to be 1.
1st method
x 3 + 5 y 3 + 2 5 z 3 = 1 5 x y z
A . M ≥ G . M
Therefore
3 x 3 + 5 y 3 + 2 5 z 3 ≥ ( 5 3 x 3 y 3 z 3 ) 3 1 )
we are given A.M = G.M
therefore x=y=z=0
2nd method
we know
x 3 + y 3 + z 3 − 3 x y z = ( x + y + z ) ( x 2 + y 2 + z 2 − x y − y z − x z )
once again by comparing we get
x=y=z=0