IMO 1964 - Problem 4

Problem: Each of 17 students talked with every other student. They all talked about three different topics. Each pair of students talked about one topic. Prove that there are three students that talked about the same topic among themselves.
Solution: Let us call the topics T1,T2,T3. Consider an aarbitrary student A. By the pigeonhole principle there is a topic, say T3, he discussed with at least 6 other students. If two of these 6 students discussed T3, then we are done.
 Suppose now that the 6 students discussed only T1 and T2 and choose one of them, say B. By the pigeonhole principle he discussed one of the topics, say T2, with three of these students. If two of these three students also discussed T2, then we are done. Otherwise, all the three students discussed only T1, which completes the task.