Question 33
Let $n \geqslant 3$ be an integer. At a MEMO-like competition, there are $3n$ participants, there are $n$ languages spoken, and each participant speaks exactly three different languages.
Prove that at least $\left\lceil\frac{2n}{9}\right\rceil$ of the spoken languages can be chosen in such a way that no participant speaks more than two of the chosen languages.
($\lceil x\rceil$ is the smallest integer which is greater than or equal to $x$.)
Prove that at least $\left\lceil\frac{2n}{9}\right\rceil$ of the spoken languages can be chosen in such a way that no participant speaks more than two of the chosen languages.
($\lceil x\rceil$ is the smallest integer which is greater than or equal to $x$.)