Question 66
For positive integers $a, b, c$ (not necessarily distinct), suppose that $a + bc$, $b + ca$, and $c + ab$ are all perfect squares. Prove that
$$
a^2(b + c) + b^2(c + a) + c^2(a + b) + 2abc
$$
can be written as the sum of two square numbers.
$$
a^2(b + c) + b^2(c + a) + c^2(a + b) + 2abc
$$
can be written as the sum of two square numbers.