Question 108
Let $a$, $b$, $c$ be positive integers. All the roots of each of the quadratics
$$
a x^{2}+b x+c,\quad a x^{2}+b x-c,\quad a x^{2}-b x+c,\quad a x^{2}-b x-c
$$
are integers. Over all triples $(a, b, c)$, find the triple with the third smallest value of $a+b+c$.
$$
a x^{2}+b x+c,\quad a x^{2}+b x-c,\quad a x^{2}-b x+c,\quad a x^{2}-b x-c
$$
are integers. Over all triples $(a, b, c)$, find the triple with the third smallest value of $a+b+c$.