Task B-3.4. Helena invited a certain number of children to her birthday party. The age of one child was one eighth of the sum of the ages of the remaining children. This age ratio did not change in later years. How many children were at the party?
## Solution.

Let $n$ be the number of children at the party, and $x$ be the age of the mentioned child.

Furthermore, let $x_{1}, x_{2}, \ldots, x_{n-1}$ be the ages of the remaining $n-1$ children. Then we have

$$
x=\frac{1}{8}\left(x_{1}+x_{2}+\cdots+x_{n-1}\right)
$$

(2 points)

If we add $t$ years to each child, the same ratio will hold, which we write as

$$
\begin{aligned}
x+t & =\frac{1}{8}\left(\left(x_{1}+t\right)+\left(x_{2}+t\right)+\cdots+\left(x_{n-1}+t\right)\right) \\
& =\frac{1}{8}\left(x_{1}+x_{2}+\cdots+x_{n-1}\right)+\frac{1}{8}(n-1) t
\end{aligned}
$$

From (1) and (2), we get $t=\frac{1}{8}(n-1) t$, and from this, $n=9$.

There were nine children at the party.