The average age of all the people who gathered for the family celebration was equal to the number of those present. Aunt Beta, who was 29 years old, soon excused herself and left. Even after Aunt Beta left, the average age of all the people present was equal to their number.

How many people were originally at the celebration?

(L. Hozová)

Hint. What is the relationship between the number of people present and the sum of their ages?
The average age of all people who gathered for the celebration is equal to the ratio of the sum of the ages of all present (denoted as $s$) and their number (denoted as $n$). According to the problem, we have

$$
\frac{s}{n}=n \quad \text { or } \quad s=n^{2}
$$

After Aunt Betty left, the number of people present decreased by 1, and the sum of their ages decreased by 29. According to the problem, we have

$$
\frac{s-29}{n-1}=n-1 \quad \text { or } \quad s-29=(n-1)^{2}
$$

When we substitute $s=n^{2}$ into the last equation, expand the right side, and further simplify, we get:

$$
\begin{aligned}
n^{2}-29 & =n^{2}-2 n+1, \\
2 n & =30, \\
n & =15 .
\end{aligned}
$$

Initially, 15 people attended the family celebration.