Question 143
Find all pairs of positive integers $(x, y)$ such that
$$
\frac{1}{x} + \frac{1}{y} + \frac{1}{\text{lcm}(x, y)} + \frac{1}{\text{gcd}(x, y)} = \frac{1}{2}.
$$
$$
\frac{1}{x} + \frac{1}{y} + \frac{1}{\text{lcm}(x, y)} + \frac{1}{\text{gcd}(x, y)} = \frac{1}{2}.
$$