<s>[INST] When the $a^2+b+c+d,b^2+a+c+d,c^2+a+b+d,d^2+a+b+c$ are all perfect squares? Find all $a,b,c,d\in \mathbb{Z}^+$, which $a^2+b+c+d,b^2+a+c+d,c^2+a+b+d,d^2+a+b+c$ are all perfect squares.
I found $(1,1,1,1)$, but I can't find more.
Is $a=b=c=d$ true?
 [/INST] Since the expressions are symmetric, then we can set $a\leq b\leq c\leq d$ without loss of generality. There is an infinite number of positive integer solutions if we assume $a=1$ and $b=c=d=n$ which yields,
$$\begin{aligned}
a^2+b+c+d &= 3n+1\\
a+b^2+c+d &= (n+1)^2\\
a+b+c^2+d &= (n+1)^2\\
a+b+c+d^2 &= (n+1)^2
\end{aligned}\tag1$$
and it is easy to find solve $3n+1 = y^2$. 
Excluding this infinite family, it seems the only solutions with all variables $1\leq a\leq b\leq c\leq d < 100$ are,
$$a,b,c,d = 6,6,11,11$$
$$a,b,c,d = 40,57,96,96$$
though I am not sure of this result. (Anyone can verify it?)
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