PLEASE I URGENTLY NEED YOUR HELP This problem is from my cla

PLEASE! I URGENTLY NEED YOUR HELP! This problem is from my class (an advanced math class):

Nine mathematicians met at an international conference. They found that among any 3 of them there are at least 2 that have a language in common. If every mathematician speaks at most 3 languages, prove that at least three of the mathematicians can speak the same language.

Solution

Let us prove this statement by contradiction:

Let us suppose that no three mathematicians(X, Y, Z) speak the same language.

Now, the person X can share some language with at most 3 other delegates, because if he shared some language with fourth delegates there would be three with the same language.

So, there are five delegates remaining who do not share language with X. Let one of them be Y.

Using the same logic, one of the remaining 4, let it be Z, who do not share language with Y.

So, it can be concluded that X, Y, Z do not have any common language.

So, we can say that at least three of the mathematecian can speak the same language.