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# help

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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
+6196
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$$\text{The only two are }\\ (1,1,1,1),~(-1,-1,-1,-1)$$

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Mar 13, 2019
#2
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I do not doubt you Rom but how do you know that those are the only 2 quadruples?

Melody  Mar 13, 2019
#3
+30620
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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
+110154
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Thanks Alan.  :)

Melody  Mar 13, 2019