Given
countries with three representatives each,
committees
are called a cycle if
(i) each committee has
members, one from each country;
(ii) no two committees have the same membership;
(iii) for
, committee
and committee
have no member in common, where
denotes
(iv) if
then committees
and
have at least one member in common.
Is it possible to have a cycle of 1990 committees with 11 countries?
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
![m](/media/m/1/3/6/1361d4850444c055a8a322281f279b39.png)
![A(1),A(2), \ldots, A(m)](/media/m/4/0/e/40e966a7bfbf4dc6b87a4f034ca1509a.png)
(i) each committee has
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
(ii) no two committees have the same membership;
(iii) for
![i = 1, 2, \ldots,m](/media/m/3/e/4/3e476dd5cad3556a64dcc6b60a63f5c6.png)
![A(i)](/media/m/a/c/a/acaada6435a652d9694cb70808b3995c.png)
![A(i + 1)](/media/m/b/2/8/b28caee5bfaba9031e70c1de7334f9a9.png)
![A(m + 1)](/media/m/7/a/3/7a3c6c8f97992fa8b6cd5c416a72bd23.png)
![A(1);](/media/m/e/a/4/ea40e1904e596bed232a34f8e8e3dec9.png)
(iv) if
![1 < |i - j| < m - 1,](/media/m/7/3/5/735fe4dad8f23bbb2b217d820c29be47.png)
![A(i)](/media/m/a/c/a/acaada6435a652d9694cb70808b3995c.png)
![A(j)](/media/m/b/e/1/be1abc40851a6b234c4ac18ef5ac0f7e.png)
Is it possible to have a cycle of 1990 committees with 11 countries?