I stuck with the following problem, which I think is relevant with pigeonhole principle.
Problem. There are $20$ translators and $18$ languages. It is known that for every given language there are exactly $5$ translators who know it. Is it true, that we can always choose $5$ translators, such that for every language there exists at least one translator knowing this language.
My attempt. We can connect every language with translators knowing it. There are $90$ arrows in our "graph". Of course there exists a translator knowing at least $5$ languages (by pigeonhole principle). After that we can remove from our graph $1$ translator and at least $5$ languages known to him. We can continue using this greedy strategy, but it is easyly to see, that it will not lead to success.
So, I will be gratefull for ideas and hints.
Assume that for $1\le i\le 6$, language $L_i$ is known by translator $T_j$ iff $1\le j\le6$ and $i\ne j$. And for $7\le i\le 12$, language $L_i$ is known by translator $T_j$ iff $7\le j\le12$ and $i\ne j$. And for $13\le i\le 18$, language $L_i$ is known by translator $T_j$ iff $13\le j\le18$ and $i\ne j$. Then each $L_i$ is known by exactly five $L_j$ (and $T_{19}$ and $T_{20}$ are not worth their money). However, in order to cover all $18$ languages, we need two translators from $T_1,\ldots,T_6$, two translators from $T_7,\ldots,T_{12}$, and two translators from $T_{13},\ldots,T_{18}$.