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수학동아 - 폴리매스
본문바로가기
함께 풀고 싶은 문제
깊이 고민해볼 수 있는 수학 문제를 내는 곳입니다
[세상을 바꿀 문제] 건망고의 취미생활
gunmgmath 2022.08.02 08:22 조회 195

건망고는 망고 사격연습을 하고 집으로 돌아와서는 자신의 취미활동을 즐긴다.

건망고는 심심할 때 다음과 같은 놀이를 한다.

1. 칠판에 n개의 1을 써 놓는다.

2. 칠판에 쓰여 있는 수들 중 임의로 2개의 수 a, b를 지우고 대신{a+b}\over4를 쓴다.

3. 위의 2. 를 칠판에 1개의 수가 남을 때까지 반복한다.

이때 건망고의 칠판에 있는 마지막 남은 수가 1\over n이상임을 보여라. 

이 문제 어떠셨나요?

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어려워요

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  •  
    해결
    굴러가던 도토리 Lv.6 2022.08.10 19:13 비밀댓글
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    •  
      Rkdals Lv.3 2022.08.12 19:17 비밀댓글
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    •  
      굴러가던 도토리 Lv.6 2022.08.12 19:37

      이 문제에서 개념이라고 말할 만한 것은 불변량/단조증가/단조감소 입니다. 조합론적인 문제에서 특정한 수를 잡았을 때, 그 수가 시행을 해도 바뀌지 않는 경우가 있고(불변량), 시행을 아무리 해도 작아지지 경우가 있고(단조증가), 시행을 아무리 해도 커지지 않는 경우가 있습니다(단조감소). 이런 수를 잡았을 때 문제를 푸는데 도움을 주는 경우가 있습니다. 이 문제처럼 말이죠.

      불변량이나 단조증가/감소량을 어떻게 찾냐 하면, 이런 문제를 많이 풀어봐야 됩니다. 불변량/단조량으로 기본적으로 시도해볼 만한 것들이 합, 차, 제곱, 곱, 역수합 등이 있습니다.

      좋아요0
    •  
      gunmgmath Lv.4 2022.08.13 01:30 비밀댓글
      비밀 댓글이 등록 되었습니다!
    •  
      Rkdals Lv.3 2022.08.17 18:28 비밀댓글
      비밀 댓글이 등록 되었습니다!
  •  
    Rkdals Lv.3 2022.08.11 21:23 비밀댓글
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