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Find the number of ordered quadruples \((a,b,c,d)\) of nonnegative real numbers such that \(\begin{align*} a^2 + b^2 + c^2 + d^2 &= 4, \\ (a + b + c + d)(a^3 + b^3 + c^3 + d^3) &= 16. \end{align*}\)

 Mar 12, 2019
 #1
avatar+6042 
+1

\(\text{The only two are }\\ (1,1,1,1),~(-1,-1,-1,-1)\)

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 Mar 13, 2019
 #2
avatar+104444 
0

I do not doubt you Rom but how do you know that those are the only 2 quadruples?

Melody  Mar 13, 2019
 #3
avatar+28159 
+2

The question stipulates non-negative numbers, so only one of Rom's sets is valid.  If the question required integer solutions then (0,0,0, 2) is also a solution.  However, it specifies real numbers (not necessarily integers), so there may well be other solutions, but so far I haven't put any thought into what those might be!

 

The only other one I can think of is (0, 0, √2, √2).

 Mar 13, 2019
edited by Alan  Mar 13, 2019
edited by Alan  Mar 13, 2019
 #4
avatar+104444 
0

Thanks Alan.  :)

Melody  Mar 13, 2019

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