Question 10
For six integers $a, b, c$ and $A, B, C$ the following correlations are true:
$$
b + c = A^2, \quad c + a = B^2, \quad a + b = C^2, \quad C \gt B \gt A \geq 0.
$$
Find numbers $a, b, c$, for which the sum $A^2 + B^2 + C^2$ takes the smallest possible value.
$$
b + c = A^2, \quad c + a = B^2, \quad a + b = C^2, \quad C \gt B \gt A \geq 0.
$$
Find numbers $a, b, c$, for which the sum $A^2 + B^2 + C^2$ takes the smallest possible value.