Question 88
a) Prove that for any real numbers $a$ and $b$ the following inequality holds:
$$
(a^2 + 1)(b^2 + 1) + 50 \ge 2(2a + 1)(3b + 1).
$$
b) Find all positive integers $n$ and $p$ such that:
$$
(n^2 + 1)(p^2 + 1) + 45 = 2(2n + 1)(3p + 1).
$$
$$
(a^2 + 1)(b^2 + 1) + 50 \ge 2(2a + 1)(3b + 1).
$$
b) Find all positive integers $n$ and $p$ such that:
$$
(n^2 + 1)(p^2 + 1) + 45 = 2(2n + 1)(3p + 1).
$$