It's nice 2
x 4 + y 4 + z 4 − 4 x y z = − 1
Find the sum of all possible values of x , y , z where x , y , z are real numbers.
- Ex. if there are two pairs of values then sum of values of x , y , z is ( 2 , 3 , 4 ) , ( 1 , 2 , − 3 ) here sum is 2 + 3 + 4 + 1 + 2 − 3
The answer is 0.
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2 solutions
Exactly....
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Ya. I loved solving this and the previous one. Conceptual :)
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well thanks.. yes i too like your way of solving :D
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@Rakshit Joshi – Ofcrs. I followed the advertisement on fb that ' come to brilliant, waste less time in fb' :P
The first line seems to be an assumption.
@Rakshit Joshi Did you intend to solve this over the integers, instead of over the reals?
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From the example he provided , It seems as if they are integers
Well it was my mistake. Well I updated it.
The problem also holds for real numbers.
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Ah thanks. I've edited the problem+solution to reflect that.
There is a simple solution to this problem that does not require finding the actual values of x, y, and z. Notice that xyz has to be positive because the sum of a positive number and -4xyz is negative. This requirement is met by having 0 or 2 of x,y,z negative. If (a,b,c) is a solution, so are (-a,-b,c), (-a,b,-c), and (a,-b,-c). More solutions may be possible by switching around the order of a, b, and c. The sum of these four possibilities (and any reorderings) is 0. Notice that this solution is valid for all real numbers, not just integers (as the question was originally stated).
Not quite. You are making the assumption that there are only finitely many solutions here, in order to conclude that the values pair up.
IE … + ( − 2 ) + ( − 1 ) + 0 + 1 + 2 + … is not an absolutely convergent sum.
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Does the fact that Brilliant is asking me for a finite solution allow me to assume that a finite solution exists?
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At the end of the day, it's up to you how much you want to shortcut and shortchange your learning. Yes, in certain settings (e.g. competitions), being able to answer a question quickly is helpful. In most other situations (e.g. learning, research), you want to know the complete reasoning and assumptions made.
You should make it explicit in your solution that "I am making the additional assumption that a numerical solution to the problem exists". IE At the extreme end, I want to avoid solutions which say "I selected the answer A and you told me I was correct, hence (I make the assumption that) I am correct".
Note that it is not true that "Brilliant problems are always completely correct, hence if they ask for a finite solution I can assume that a finite solution exists". That is why we have reports which allow people to point out errors.
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@Calvin Lin – Ok, thank you for pointing that out.
By arithmetic mean - geometric mean ,
x 4 + y 4 + z 4 + 1 ≥ 4 x y z
equality holds iff x 4 = y 4 = z 4 = 1
We may have any two of them negetive but not all three as x 4 + y 4 + z 4 > 0 i.e 4 x y z > 1
So we only have the solutions (1,1,1) , (-1,-1,1),(-1,1,-1),(1,-1,-1) .
So adding all the values of x,y,z we get 0