Arrangements around a circle $5$ mathematicians, $5$ biologists, $5$ chemists, $5$ physicists, and $5$ economists sit around a large round table. Prove that the $25$ people can be seated such that, if $A$ and $B$ are two different people with the same specialty (for example, two mathematicians), then the people sitting to the immediate left of $A$ and to the immediate left of $B$ are of different specialties (for example, a biologist and a chemist).

Should I try to use induction? 
 A: This is a slightly easier version of problem $1$ of the second day of the 1998 Iberoamerican Olimpiad available here.
We can solve it by induction. We prove the statement, if there are $n$ science fields and $n$ scientists from each field we can sit them so that that if $A$ and $B$ belong to the same field then the person to the left of $A$ and the person to the left of $B$ belong to different fields.
Base case: (I trust you can find suitable arrangements for $n=1$ and $2$.
Inductive step. We take a suitable arrangement for $n$ people. We must add $n$ people belonging to the new field and one person from each of the old $n$ fields of science. To do this choose any two seats in the old arrangement and introduce the $2n$ new empty seats in between them. Suppose the new field of science is Wolframology, We fill the $2n$ new fields of science by placing a Wolframologist in the first place, then a scientist from a diferent field , then a wolframologist, and so on. Alternating between a Wolframologist and one of the old sciences. If the two seats we picked initially where of sciences $A$ and $B$ just make sure that the final seat picked is occupied by a scientist of science $A$. This arrangement works because al of the Wolframologists have scientists from different fields to their left and all of the other new scientists have Wolframologists to their left.
You can find other solutions to the problem in the link that I purveyed.
