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

Guest Mar 12, 2019

#3**+1 **

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).

Alan Mar 13, 2019