Question 56
Determine the greatest possible value of $M$ for which:
$$
\frac{x}{1+\frac{yz}{x}} + \frac{y}{1+\frac{zx}{y}} + \frac{z}{1+\frac{xy}{z}} \ge M,
$$
for all real numbers $x, y, z \gt 0$ satisfying the equation: $xy + yz + zx = 1$.
$$
\frac{x}{1+\frac{yz}{x}} + \frac{y}{1+\frac{zx}{y}} + \frac{z}{1+\frac{xy}{z}} \ge M,
$$
for all real numbers $x, y, z \gt 0$ satisfying the equation: $xy + yz + zx = 1$.