5. Let $x, y, z$ be the roots of the equation $t^{3}-2 t^{2}-9 t-1=0$. Find $\frac{y z}{x}+\frac{x z}{y}+\frac{x y}{z}$.

(12 points)
Solution. Let's bring the desired expression to a common denominator: $\frac{y^{2} z^{2}+x^{2} z^{2}+x^{2} y^{2}}{x y z}$. The polynomial has 3 different real roots, since $\mathrm{P}(-100)0, \mathrm{P}(0)0$. By Vieta's theorem $x+y+z=2, x y+x z+y z=-9, x y z=1$.

$$
\begin{aligned}
& x^{2} y^{2}+x^{2} z^{2}+y^{2} z^{2}=(x y+x z+y z)^{2}-2\left(x^{2} y z+y^{2} x z+z^{2} x y\right) \\
& =(x y+x z+y z)^{2}-2 x y z(x+y+z)=81-2 * 1 * 2=77
\end{aligned}
$$

Answer: 77.