Danka had a paper flower with ten petals. On each petal, there was exactly one digit, and no digit was repeated on any other petal. Danka tore off two petals so that the sum of the numbers on the remaining petals was a multiple of nine. Then she tore off another two petals so that the sum of the numbers on the remaining petals was a multiple of eight. Finally, she tore off two more petals so that the sum of the numbers on the remaining petals was a multiple of ten.

Find the three sums that could have remained after each tearing. Determine all such triplets of sums.

(E. Novotná)
The sum of all numbers on the petals is $0+1+\ldots+9=45$. Danka could have torn off a sum of at least 1 (if she tore off petals 0 and 1) and at most 17 (if she tore off petals 8 and 9).

- After the first tearing, the sum remaining on the flower must be in the range of 28 to 44 ( $45-17=28$ and $45-1=44$ ). Among these numbers, there is only one multiple of nine, which is 36.
- After the second tearing, the sum remaining on the flower must be in the range of 19 to 35 ( $36-17=19$ and $36-1=35$ ). Among these numbers, there are two multiples of eight, namely 32 or 24.
- Similarly, we determine the possible sums after the third tearing:
- If the sum remaining after the second tearing is 32, the sum remaining after the third tearing must be in the range of 15 to 31 ( $32-17=15$ and $32-1=31$ ). Among these numbers, there are two multiples of ten, namely 30 or 20.
- If the sum remaining after the second tearing is 24, the sum remaining after the third tearing must be in the range of 7 to 23 ( $24-17=7$ and $24-1=23$ ). Among these numbers, there are two multiples of ten, namely 20 or 10.

After each tearing, the following triples of sums could remain:

$$
(36,32,20),(36,32,30),(36,24,20),(36,24,10).
$$

For each of these possibilities, we need to verify whether they can actually be realized, i.e., whether pairs of petals can be torn off in such a way that no digit is used twice. In the following scheme, we write above the arrows an example of which petals could have been torn off in each step:

$$
\begin{aligned}
& 45 \xrightarrow{0,9} 36 \xrightarrow{1,3} 32 \xrightarrow{4,8} 20 \\
& 45 \xrightarrow{4,5} 36 \xrightarrow{1,3} 32 \xrightarrow{0,2} 30 \\
& 45 \xrightarrow{2,7} 36 \xrightarrow{4,8} 24 \xrightarrow{1,3} 20 \\
& 45 \xrightarrow{2,7} 36 \xrightarrow{4,8} 24 \xrightarrow{5,9} 10
\end{aligned}
$$

We see that all the given possibilities can be realized, the problem has four solutions.

Evaluation. 2 points for finding the four possibilities; 2 points for verifying that each possibility can be realized; 2 points for a process that excludes the possibility of missing any other possibility, or for justifying that no other possibility exists.