3. If we divide the whole positive number $N$ by the number 19 and then divide the obtained incomplete quotient by the number 99, the incomplete quotient and the remainder of the second division are the same as when the original number $N$ is divided by 1999. Determine the smallest and the largest such number $N$.

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

on Tuesday, January 25, 2000

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. The mentioned three divisions can be written as equations $N=19a+b$, $a=99c+d$, and $N=1999c+d$. From these, it follows that $19(99c+d)+b=1999c+d$, or $18d+b=118c$. The smallest and largest suitable $N$ can be found according to the smallest and largest possible quotient $c$ (when dividing the number $N$ by 1999). From the equation $18d+b=118c$, it follows primarily that $c>0$ (if $c=0$, then $b=d=0$, and thus $N=0$, but $N$ is positive); for $c=1$, from the equation $18d+b=118$, we conclude that $d=6$ and $b=10$ (since for the remainder $b$ when dividing $N$ by 19, $0 \leq b \leq 18$). Therefore, the smallest $N$ is equal to the number $1999 \cdot 1 + 6 = 2005$. To find the largest $N$, note first that for the remainders $d$ and $b$ when dividing $a$ by 99 and $N$ by 19, the inequalities $d \leq 98$ and $b \leq 18$ hold, from which we get the estimate $118c \leq 18 \cdot 98 + 18$, hence $c \leq 15$. For $c=15$, however, from the equation $18d+b=118 \cdot 15$, it follows that $d=98$ and $b=6$, since $0 \leq b \leq 18$ and $118 \cdot 15 = 1770 = 18 \cdot 98 + 6$. The largest $N$ is thus equal to $1999 \cdot 15 + 98 = 30083$.

Answer: The smallest suitable $N$ is 2005, and the largest such $N$ is 30083.

For a complete solution, award 6 points, of which 2 points for finding one of the two numbers, and an additional 1 point for the justification of why it is indeed the extreme value.