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수학동아 - 폴리매스
함께 풀고 싶은 문제
창의력을 기를 수 있는 수학 문제 또는 퍼즐을 내는 곳입니다.
[창의 퍼즐] [조합] 코로나 시국에 악수를 한다?
△π
2020.09.02 19:01
조회 502
총 n명의 외교관들이 모여 있다. 임의의 네 외교관 A,B,C,D에 대하여, A와 B가 악수를 했고 B와 C가 악수를 했으며 C와 D가 악수를 했다면, 세 쌍 A와 C, A와 D, B와 D 중 악수를 했던 쌍이 반드시 존재한다. 이 때 다음을 증명하여라.
(a) 전체 외교관을 다음 성질이 만족하도록 공집합이 아닌 두 집합 X,Y로 나눌 수 있다. X에 속한 모든 외교관이 Y에 속한 어떤 외교관과도 악수를 하지 않았거나, X에 속한 모든 외교관이 Y에 속한 모든 외교관과 악수를 하였다.
(b) 어떤 두 외교관 A,B가 있어서 A,B 이외의 외교관 중에 A와 악수한 사람들의 모임과 B와 악수한 사람들의 모임이 같다.
(단, n은 4 이상이다.)
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