It shows the level of sophistication in IMO in the '70ties Today this is classroom stuff. Out of the $n \geq k^3 + k$ even numbers in $M$, at least $\dfrac {k^3 + k} {k} = k^2 + 1$ will be in a same $M_i$, by pigeonhole principle, or what I call an averaging argument. Then at least $k+1$ of those will have their odd predecessors in a same $M_j$ (with maybe $i=j$), by the same type of argument. Moreover, notice it is enough to have $n\geq k^3 + 1$, since again it will exist some $M_i$ holding at least $k^2+1$ even numbers from $M$, etc.