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The answer is $n=20$. First of all, clearly if $n<20$ it is impossible. Pick a specific country, say country $A$, and then, for each language that country $A$ does not know add a person that does not know that language to the group, by doing this you can create a group of $20$ persons such that none no language is common to all of them. With $n=20$ it is possible, pick a ground set of $21$ languages and make sure that every country can speak $20$ of those languages (so that there is at least one country that does not speack every specific language). Notice that the fact that there are $100$ countries is irrelevant.

Gamamal wrote: The answer is $n=20$. First of all, clearly if $n<20$ it is impossible. Pick a specific country, say country $A$, and then, for each language that country $A$ does not know add a person that does not know that language to the group, by doing this you can create a group of $20$ persons such that none no language is common to all of them. With $n=20$ it is possible, pick a ground set of $21$ languages and make sure that every country can speak $20$ of those languages (so that there is at least one country that does not speack every specific language). Notice that the fact that there are $100$ countries is irrelevant. With n< 20 , why is it impossible ?

Infact its possible with even n=2 ; I am quite sure ..I can even explain you how .

I think the flaw in reasoning rests in the justification for why $n < 20$ is said to be impossible. This construction is assuming a worst-case scenario with minimizing language knowledge overlap that may not hold. Obviously we cannot have $n = 1$, so it would only remain to show that there is an explicit construction for $n = 2$ (or, perhaps at a higher difficulty level, use an abstract existence proof, I'm not sure which is easier...).

If you say that 99 countries have a common language , and only one country doesn't know that language , then whatever 20 set countries form with that country can communicate with that language and so you can have n=2 .

For sake of completeness, we should add that any combination of 20 out of the first 99 countries can communicate with the other language. Duh that was obvious. Also, 2222nd post woo

Kingofmath101 wrote: For sake of completeness, we should add that any combination of 20 out of the first 99 countries can communicate with the other language. Duh that was obvious. Also, 2222nd post woo Yup that is quite clear ...

Kk108...n=2 ....is impossible....ans is 20...

svatejas wrote: There are \(100\) countries participating in an olympiad. Suppose \(n\) is a positive integers such that each of the \(100\) countries is willing to communicate in exactly \(n\) languages. If each set of \(20\) countries can communicate in at least one common language, and no language is common to all \(100\) countries, what is the minimum possible value of \(n\)? Sorry for reviving but please be careful while posting problems, especially in contest collections. The words "exactly" and "at least" create a big difference in the meaning.