## Task 1 - 340831

On 10 cards, the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 are written, each digit on exactly one card. Anna selects three of these cards and lays them out on the table twice, the first time as the representation of the largest number, and the second time as the representation of the second-largest number that can be achieved with these three cards.

Anna reports: The sum of the two numbers, whose representations she laid out, is 1233. Determine all possibilities for which two numbers Anna could have laid out!
The digits on the cards chosen by Anna are denoted by $a, b, c$ such that $a > b > c$. Anna has placed the numbers $100a + 10b + c$ and $100a + 10c + b$. Therefore, Anna's statement is true if and only if these digits satisfy the equation

$$
200a + 11 \cdot (b + c) = 1233
$$

I. If (1) is satisfied, it follows: Since $0 \leq 5$ and $1222: 200c$ can only be satisfied if either $b = 3, c = 0$ or $b = 2, c = 1$ (3) holds.

II. For both possibilities mentioned in (2), (3) for $a, b, c$, (1) is satisfied.

With I. and II., it is shown: There are exactly two possibilities, that Anna either placed the two numbers 630, 603 or the two numbers 621, 612.