Question 73
Let $a$ and $b$ be positive numbers. Find all pairs of functions $f, g: \mathbf{R} \to \mathbf{R}$, each assuming the value $1$ and fulfilling, for any $y \ne 0$ and any $x$, the equations
$$
f\left(\frac{1}{y^2}g(xy) - ax^2\right) = 0 = g\left(\frac{1}{y}f(xy) - bx\right).
$$
$$
f\left(\frac{1}{y^2}g(xy) - ax^2\right) = 0 = g\left(\frac{1}{y}f(xy) - bx\right).
$$