Question 59
Let $a$, $b$, $c$ be distinct positive integers.
a) Prove that $a^2b^2 + a^2c^2 + b^2c^2 \ge 9$.
b) If, moreover, $ab + ac + bc + 3 = abc \gt 0$, show that
$$
(a - 1)(b - 1) + (a - 1)(c - 1) + (b - 1)(c - 1) \ge 6.
$$
a) Prove that $a^2b^2 + a^2c^2 + b^2c^2 \ge 9$.
b) If, moreover, $ab + ac + bc + 3 = abc \gt 0$, show that
$$
(a - 1)(b - 1) + (a - 1)(c - 1) + (b - 1)(c - 1) \ge 6.
$$