A kind of Ramsey theroy question... Here is my silly idea which I use today:Consider an alien $O_1$,there exist an language $L_1$ which $O_1$ communicates to at least $3\times2004!$ aliens (let those aliens as Group A)with $L_1$(Easy to prove by Pigeonhole Principle),if any two aliens in Group A communicates to each other with$L_1$,then that two aliens and $O_1$ communicate with each other in one common language$L_1$. Else among Group A,they can use the other $2004$ languages to communicates to each other only.Consider alien$O_2$ in Group A,there exist an language $L_2$ which $O_2$ communicates to at least $3\times2003!$ aliens in Group A with $L_2$ (let those aliens as Group B),if any two aliens in Group B communicates to each other with$L_2$,then that two aliens and $O_2$ communicate with each other in one common language$L_2$. Repeat this logic,until the case"among $3$ aliens,they can only use $1$ language to communicates to each other",which is trivial...

Everyone applied induction on this problem. I wonder if there is any other ways to tackle it.

hy i wonder what were the results of the selection test? how many people solved 3 or more problems?

This problem is in the list of 2005 HK TST but the title of this suggests that it was 2006 HK TST. Why is that?