## Task B-2.3.

Dora, Magdalena, and Luka are playing a "word game." Dora started the game by saying one word. Magdalena repeated what Dora said and added a new word. Luka repeated the words that Dora and Magdalena had said and added a new word. They continued like this, with Dora, Magdalena, and Luka taking turns, until one person made a mistake. That person, when it was their turn to play, successfully repeated 10 words but then forgot the next one and did not continue the sentence. Determine who made the mistake and how many words they should have correctly repeated to continue the game, given that the total number of words correctly spoken during this game was 4960.
## Solution.

Let $n$ be the number of words the person who made a mistake was supposed to say (repeat $n-1$ words and add one new one).

Dora, Magdalena, and Luka each correctly said $1, 2, \ldots, n-1$ words before someone made a mistake and said 10 out of the total $n$ words they were supposed to say for the game to continue.

Then the total number of words is $1+2+3+\cdots+n-1+10=4960$.

From this, we have

$$
\begin{gathered}
\frac{(1+n-1)(n-1)}{2}-4950=0 \\
n^{2}-n-9900=0 \\
n^{2}-100 n+99 n-9900=0 \\
(n-100)(n+99)=0
\end{gathered}
$$

The positive solution to this equation is $n=100$.

Therefore, the person who made a mistake was supposed to repeat exactly 99 words, or say a total of 100 words.

Since $100=33 \cdot 3+1$, we conclude that Dora is the person who made the mistake.