Question 39
Points $A$ and $B$ and a circle $\mathcal{K}$ with diameter $AB$ are given. On the segment $AB$ we choose a point $T$ different from $A$ and $B$. Let the perpendicular to the segment $AB$ through the point $T$ intersect the circle $\mathcal{K}$ at points $M$ and $N$. Denote $|AT|=x$, $|TB|=y$ and $|TN|=z$. Compute the value of the expression
$$
\frac{\log_{y} z + \log_{x} z}{\log_{x} z \log_{y} z}
$$
$$
\frac{\log_{y} z + \log_{x} z}{\log_{x} z \log_{y} z}
$$