Insert two digits into the number 2019 so that the resulting six-digit number

- starts with 2 and ends with 9,
- is composed of six different digits,
- is divisible by three,
- its first three-digit number is divisible by three,
- its first four-digit number is divisible by four,
- the sum of the inserted digits is odd.

Determine the difference between the largest and smallest six-digit number with the given properties.

(L. Růžičková)
To form a new number consisting of six different digits, we can insert two different digits from the digits

$$
3,4,5,6,7,8
$$

The sum of the digits of the number 2019 is 12, which means the number is divisible by three. To ensure the newly formed number is also divisible by three, we can only insert digits whose sum is divisible by three. The last condition in the problem also requires this sum to be odd. From all possible pairs of numbers, we can only use the following (in any order):

$$
3,6, \quad 4,5, \quad 7,8
$$

To ensure the new number starts with 2 and ends with 9, we can only insert digits in the positions marked with an asterisk:

$$
2 * * 019, \quad 2 * 0 * 19, \quad 2 * 01 * 9, \quad 20 * * 19, \quad 20 * 1 * 9, \quad 201 * * 9 .
$$

To ensure the first three-digit number is divisible by three, we cannot add any of the mentioned pairs in the first case. In the second to fifth cases, we can add the pairs 4,5 or 7,8 (in this order), and in the last case, we can add any of the mentioned pairs (in any order):

$$
\begin{array}{llllll}
2 \underline{4} 0 \underline{5} 19, & 2 \underline{7} 0 \underline{8} 19, & 2 \underline{4} 01 \underline{5} 9, & 2 \underline{7018} 9, & 20 \underline{4} 1 \underline{5} 9, & 20 \underline{71} \underline{8} 9, \\
201369, & 201 \underline{639} 9, & 201 \underline{45} 9, & 201549, & 201789, & 201 \underline{87} 9 .
\end{array}
$$

To ensure the first four-digit number is divisible by four, the second two-digit number must be divisible by four. From the mentioned possibilities, only two remain:

$$
2 \underline{7} 0 \underline{8} 19, \quad 201 \underline{63} 9 \text{. }
$$

We have used all the requirements from the problem; the requested difference is 69180.

Evaluation. 3 points for the correct options and the final difference; 3 points for the quality, or completeness, of the commentary.