341I think n = 8. Bit of a clumsy proof though
Obviously n>5.
Now represent the participants by dots, so as to form the vertices of a polygon. When we form a triangle using three vertices, we can label it by an li for one of the 14 languages spoken.
If n = 6, we have 6C3 = 20 triangles. Since each language can have up to three speakers, every triangle has to be labelled by a distinct index. But we have only 14 indices available
The same reasoning excludes n = 7.
For n =8, we can have up to 4 speakers per language. Now we have 56 triangles. Suppose you label the vertices 1,2,..,8, then we find fourteen quadrilaterals that have not more than 1 edge common as below:
1 2 3 4
1 2 5 6
1 2 7 8
3 4 5 6
3 4 7 8
5 6 7 8
1 3 6 8
1 3 5 7
2 4 5 7
2 4 6 8
1 4 5 8
1 4 6 7
2 3 5 8
2 3 6 7
Each of these 14 quadrilaterals has four triangles which are not shared with any other triangle (since no two quadrilaterals share more than one edge). Thus the 56 triangles are partitioned into 14 groups of four such that each group has 4 vertices.
Label each of these 14 groups by Q1 , Q2,....Q14. In each Qi all the four triangles are assigned li
Its now easy to see that we have met the conditions as
(1) every triangle has been labelled and hence any three speakers have a language in common
(2) All the triangles that are labelled by the same index are formed from the four vertices of a quadrilateral, so that each language is spoken by exactly four speakers.
And so the minimum number of participants is 8
[I am sure there is a more elegant way of presenting this. So I would like to see a simpler solution]
