3. The quadratic equation $x^{2}-35 x+334=0$, whose coefficients are written in a number system with base $z(z \geqq 6)$, has two distinct real roots. Determine $z$ and both roots.

School - the written part of the 1st round of category B takes place

## on Tuesday, January 26, 1999

so that it starts in the morning and the contestants have 4 hours of pure time to solve the problems. For each problem, a contestant can earn 6 points, and a successful solver is a student who scores 10 points or more. These details will be communicated to the students before the start of the competition.
## 3. Given equation

$$
x^{2}-(3 z+5) x+\left(3 z^{2}+3 z+4\right)=0
$$

has two distinct real roots if and only if its discriminant $D$ is positive,

$$
\begin{aligned}
D & =(3 z+5)^{2}-4\left(3 z^{2}+3 z+4\right)=-3 z^{2}+18 z+9= \\
& =-3\left(z^{2}-6 z-3\right)=-3\left((z-3)^{2}-12\right)>0
\end{aligned}
$$

from which $z0 \mathrm{v}$ in the domain of integers $z \geqq 6,1$ point for calculating the roots $x_{1,2} \mathrm{v}$ in the case $z=6$.