The mumbo-jumbo tribe's language consists of $k$ different words. The words only contain two letters, $A$ and $O$. No word can be obtained from another by removing some letters from the end. What is the minimum number of letters contained in the complete dictionary of the language?
Let's say a language is a prefix if it has the property mentioned in the task. Suppose that with fewer letters, a complete dictionary of a $k$-word prefix language cannot be formed in the Mumbo-Jumbo root language. A sequence made of $A$s and $O$s is free if it cannot be obtained by omitting some letters from the end of a Mumbo-Jumbo word. Let's denote the length of the shortest free sequence by $h$ and the length of the longest word by $H$. Since the words of the language are free sequences, $h \leq H$.

Let $p$ be any $H$-letter word, and $\bar{p}$ the sequence obtained from $p$ by replacing its last letter with the other letter. If $\bar{p}$ were not a word, we could omit the last letter of $p$, and the language would remain a prefix. However, this would contradict our assumption that a prefix dictionary cannot be formed with fewer letters, so $\bar{p}$ must be a word.

Suppose first that $h=H$. Then every $H$-element sequence is a word, otherwise, by omitting the last letter, we would get a free sequence. Thus $k=2^{H}$, and the complete dictionary consists of $k H$ letters.

We will show that if $h<H$, then $h$ can only be $(H-1)$. Let's denote the number of $h$-element free sequences by $s$. From the other $h$-element sequences, by appending the letters $A$ and $O$, we get $2-2 H$ letter words. Therefore,

$$
k=s+2\left(2^{h}-s\right)=2^{H}-s
$$

and the complete dictionary

$$
h s+H\left(2^{h}-s\right)=k H-s
$$

consists of letters, summarizing our results, we can say that if $2^{H}$ is the smallest power of 2 that is not less than $k$, then the Mumbo-Jumbo language dictionary consists of at least

$$
k H+\left(2^{H}-k\right)
$$

letters.

Remark. It is easy to explicitly give the minimal dictionary. In our solution, this was not necessary, since without it, we knew that there is a minimal dictionary, and we were able to determine the number of letters in it unambiguously without knowing the specific form of the minimal dictionary.