IMO 1975 SL 1

Origin: FRA

Problem

There are six ports on a lake. Is it possible to organize a series of routes satisfying the following conditions: (i) Every route includes exactly three ports; (ii) No two routes contain the same three ports; (iii) The series offers exactly two routes to each tourist who desires to visit two different arbitrary ports?

Solution

First, we observe that there cannot exist three routes of the form (A, B, C), (A, B, D), (A, C, D), for if E, F are the remaining two ports, there can be only one route covering A, E, namely, (A, E, F). Thus if (A, B, C), (A, B, D) are two routes, the one covering A, C must be w.l.o.g. (A, C, E). The other roots are uniquely determined: These are (A, D, F), (A, E, F), (B, D, E), (B, E, F), (B, C, F), (C, D, E), (C, D, F).