4B. Three cards are given. The number 19 is written on one, the number 97 on another, and a two-digit number on the third. If we add all the six-digit numbers obtained by arranging the cards in a row, we get the number 3232320. What number is written on the third card?
Solution. First method. Let $x$ be the two-digit number written on the third card. There are six possibilities for arranging the three cards in a row, and the following numbers are obtained:

$$
\overline{1997 x}, \overline{19 x 97}, \overline{x 1997}, \overline{9719 x}, \overline{97 x 19}, \overline{x 9719}
$$

For their sum, we get:

$$
\begin{aligned}
S & =2 \cdot 19(10000+100+1)+2 \cdot 97(10000+100+1)+2 x(10000+100+1) \\
& =20202(19+97+x)
\end{aligned}
$$

From the condition $S=3232320$ we find:

$$
19+97+x=160 \Rightarrow x=44
$$

Therefore, the number 44 is written on the third card.

Second method. Let $\overline{a b}$ be the two-digit number written on the third card. There are six possibilities for arranging the three cards in a row, and the following numbers are obtained:

$\overline{1997 a b}, \overline{19 a b 97}, \overline{a b 1997}, \overline{9719 a b}, \overline{97 a b 19}, \overline{a b 9719}$.

For their sum, we get:

$$
\begin{aligned}
& S=1 \cdot 202020+9 \cdot 222222+7 \cdot 20202+a \cdot 202020+b \cdot 20202 \\
& 2343432+a \cdot 202020+b \cdot 20202=3232320 \\
& (10 a+b) \cdot 20202=888888 \\
& 10 a+b=44 \\
& \overline{a b}=44
\end{aligned}
$$

Therefore, the number 44 is written on the third card.

## Second Year

1AB. For which values of the parameter $a$, does the system of linear equations

$$
\left\{\begin{array}{l}
3 x+n y=3 \\
2 x-4 y=1
\end{array}\right.
$$

have a solution that satisfies the conditions $x>0$ and $y<0$?