## Task 1 - 310731

At a birthday party, candies were distributed to the children:

The first child received 1 candy and one-tenth of the remaining,

The second child received 2 candies and one-tenth of the now remaining,

The third child received 3 candies and one-tenth of the now remaining, etc.

In the end, when this was consistently continued, all the candies were distributed, and it turned out that each child had received the same number of candies as every other child.

Determine from these data the total number $a$ of all distributed candies, the number $k$ of all participating children, and the number $b$ of candies that each of these children received!

Verify that for the numbers $a, k, b$ you have determined, all the above statements are true!
From the given information: The 1st child received 1 + $\frac{1}{10}(a-1)=\frac{a}{10}+\frac{9}{10}$ candies, after which there were $a-\left(\frac{a}{10}-\frac{9}{10}\right)=$ $\frac{9 a}{10}-\frac{9}{10}$ candies left.

The 2nd child received $2+\frac{1}{10}\left(\frac{9 a}{10}-\frac{9}{10}-2\right)=\frac{9 a}{100}+\frac{171}{100}$ candies.

Since the 1st child also received this amount, it follows that

$$
\frac{9 a}{10}-\frac{9}{10}=\frac{9 a}{100}+\frac{171}{100} \quad \Rightarrow \quad a=81
$$

and thus further

$$
b=\frac{a}{10}+\frac{9}{10}=9
$$

Since each of the $k$ children received $b$ candies, the total number of distributed candies is $a=k \cdot b$; thus, $k=81: 9=9$.

Verification:

| Number of the child | Number of candies given to this child | Remaining after distribution |
| :---: | :---: | :---: |
| 1 | $1+80: 10=9$ | $81-9=72$ |
| 2 | $2+70: 10=9$ | $72-9=63$ |
| $\ldots$ | $\ldots$ | $\ldots$ |
| 8 | $8+10: 10=9$ | $18-9=9$ |
| 9 | $9+0: 10=9$ | $9-9=0$ |