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폴리매스 문제
아직 아무도 답을 모르는 문제에 도전하세요!
[대한수학회] 대41. 유리수점? 정수점? 곱가능집합?
수학동아 2020.05.01 09:11 조회 2873

대한수학회 41번

 

유리수점? 정수점? 곱가능집합?

 

 

문제 출제자 : 오병권 서울대학교 수리과학부 교수

 

 

좌표평면 상의 점 (p, q)에 대해 p, q가 모두 유리수이면 (p, q)유리수점이라 정의하고,

 

p, q가 모두 정수이면 (p, q)정수점이라 정의한다.

 

 

좌표평면 상의 점 (p, q), (r, s)와 임의의 실수 t에 대해

 

(p, q)+(r, s)=(p+r, q+s),       t(p, q)=(tp, tq)

 

로 각각 정의하자.

 

 

좌표평면 상의 점 (a_1, b_1), (a_2, b_2), \cdots , (a_r, b_r)에 대해

 

S_{(a_1, b_1), (a_2, b_2), \cdots , (a_r, b_r)}={x_1(a_1, b_1)+x_2(a_2, b_2)+\cdots +x_r(a_r, b_r) :x_1,\cdots ,x_r은 정수}

 

라 정의하자. 또한

 

LS_{(a_1, b_1), (a_2, b_2), \cdots , (a_r, b_r)}=\left \{ \sqrt{x^2+y^2} : (x, y)\in S_{(a_1, b_1), (a_2, b_2), \cdots , (a_r, b_r)} \right \}

 

라 정의하자.

 

 

실수의 부분집합 E가, 모든 x, y \in E에 대해 xy \in E를 만족하면 곱가능집합이라고 정의한다. 예를 들어, LS_{(1, 0), (0, 1)}은 곱가능집합이고 LS_{(1, 1), (0, 3)}은 곱가능집합이 아님을 쉽게 확인할 수 있다.

 

 

 

문제1 만약 (a_1, b_1), (a_2, b_2), \cdots , (a_r, b_r)이 모두 유리수 점이면 S_{(a_1, b_1), (a_2, b_2), \cdots , (a_r, b_r)}=S_{(p, q), (r, s)}를 만족하는 유리수 점 (p, q), (r, s)가 존재함을 보여라.

 

 

문제2 임의의 정수점 (p, q)에 대해, LS_{(1, 0), (p, q)}는 곱가능집합임을 보여라. 

 

 

문제3 정수 m3m^2-1을 나누는 소수 p에 대해, LS_{(1, m), (0, p)}는 곱가능집합임을 보여라.

 

 

문제4 집합 LS_{(p, q), (r, s)}가 곱가능집합이 되는 정수 p, q, r, s를 모두 구하라.

  •  
    해결
    구머 Lv.6 2020.05.01 10:51 비밀댓글

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    비밀 댓글이 등록 되었습니다.
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    •  
      구머 Lv.6 2020.05.01 10:56

      2번풀이입니다. 비교적 난이도가 쉬운 애라 많이 도전해보시길!

      좋아요0
    •  
      MathlabJ Lv.8 2020.05.01 17:59

      구머님 어떻게 푸셨나요.. 문제가 오늘 올라왔는데 어떻게 이렇게 빨리 푸셨지.....ㄷㄷㄷㄷ

      좋아요0
    •  
      여백 패르마 Lv.5 2020.05.01 19:17

      혹시 라그랑주 항등식 사용하셨나요? 

      좋아요0
    •  
      구머 Lv.6 2020.05.01 19:58

      좋아요0
    •  
      여백 패르마 Lv.5 2020.05.01 20:55

      어떻게 푸셨는지 감이 오네요

      좋아요0
    •  
      최기자 Lv.4 2020.07.17 07:13

      김다인 멘토의 검토 결과 2번을 잘 풀었습니다. 피드백이 늦어서 미안합니다~!

      좋아요0
  •  
    21세기오일러 Lv.11 2020.05.01 20:00

    이번문제는 몇 안되는 제가 이해한 대한수학회문제네요. (요새 약간 난이도 하향된 느낌?)

    댓글 작성하기 좋아요0 댓글수1
    •  
      21세기오일러 Lv.11 2020.05.01 22:18

      (내가 이해를 못하는 문제= 다른 분들은 이해가능, 내가 이해를 하는 문제= 다른 분들은 바로 풀어버림 ㅋㅋ)

      좋아요0
  •  
    여백 패르마 Lv.5 2020.05.01 20:45

    3번에서 소수 p가 1이기 때문에 모순 아닌가요? m과 3m^2-1은 반드시 서로소이잖아요

    댓글 작성하기 좋아요0 댓글수3
    •  
      구머 Lv.6 2020.05.02 09:39

      그러게요 확인 부탁드립니다 출제자님

      좋아요0
    •  
      여백 패르마 Lv.5 2020.05.03 04:09

      꼭 확인해주세욧!!!

      좋아요0
    •  
      최기자 Lv.4 2020.05.07 00:06

      안녕하세요.^^ 답이 늦어서 미안합니다~!

       

      문제를 해석하는 데 약간의 오류가 있는 것 같아요.

       

      m과 소수 p를 뜻합니다. 이 소수가 3m^2-1을 나눈다는 뜻입니다. m은 나누든 말든 아무 상관없구요.

       

      정수 m이 주어져 있고, 3m^2-1을 나누는 소수 p를 잡아... 이런 뜻입니다.^^

      좋아요0
  •  
    유한의끝도못본남자 Lv.7 2020.05.01 21:07

    위치벡터로 풀면되겠네요 이번건 좀 쉽네요

    댓글 작성하기 좋아요0 댓글수3
    •  
      구머 Lv.6 2020.05.01 22:51

      좋아요0
    •  
      여백 패르마 Lv.5 2020.05.01 22:56

      저만 위치벡터로 어떻게 푸는지 모르나봅니다...

      좋아요0
    •  
      은총알 Lv.11 2020.05.22 07:20

      저만 위치벡터가 뭔지 모르나봅니다..

      좋아요0
  •  
    다시 도전
    유한의끝도못본남자 Lv.7 2020.05.01 21:42 비밀댓글

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    •  
      유한의끝도못본남자 Lv.7 2020.05.01 21:44

      3번풀이입니다

      좋아요0
    •  
      유한의끝도못본남자 Lv.7 2020.05.01 21:45

      집합을 잘못작성했지만, 임의의모든 x_1과 뜻이 같으니 풀이로서는 맞습니다

      좋아요0
    •  
      최기자 Lv.4 2020.07.17 07:15

      안녕하세요. 김다인 멘토가 검토해서 다음과 같은 피드백을 주었습니다.

       

      “n=정수, p=소수이므로 x_1, x_2가 유리수일때는 자명하다” 부분에서 추가적인 논의를 해주세요! 특히, 풀이에서 p가 3m^2-1을 나눈다는 성질이 안 쓰였다는 점에서 다시 한 번 생각해볼 필요가 있겠죠? [1번] 마찬가지로 “수학적인” 논의 부탁드립니다.

      좋아요0
  •  
    유한의끝도못본남자 Lv.7 2020.05.01 21:47 비밀댓글
    확인요청중
    비밀 댓글이 등록 되었습니다.
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    •  
      수학꿀잼 Lv.3 2020.05.07 03:51

      혹시 몇번인가요?

      좋아요0
  •  
    파스칼 Lv.7 2020.05.02 07:11 비밀댓글
    확인요청중

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    비밀 댓글이 등록 되었습니다.
    댓글 작성하기 댓글수2
    •  
      파스칼 Lv.7 2020.05.02 08:29

      1번 풀이입니다.

      좋아요0
    •  
      최기자 Lv.4 2020.07.17 07:16

      확인이 늦어서 미안합니다. 김다인 멘토가 검토한 뒤 다음과 같은 의견을 줬어요^^

       

      시각화해서 문제에 접근한 점이 매우 좋습니다! 다만 시각화의 단점이 수학적인 허점이 쉽게 보이지 않는다는 거예요. (a_1, b_1) = (1,0), (a_2, b_2) = (2,0)이라고 해볼까요? 만약 (a_3, b_3) = (0,1)이라면 n(a_3, b_3)는 임의의 0이 아닌 정수 n에 대해서 S_{(a_1, b_1), (a_2, b_2)} 안에 속하지 않겠죠? 사소한 부분들을 조금 더 다듬으면 좋을 것 같네요!
       

      좋아요0
  •  
    구머 Lv.6 2020.05.02 09:17

    1번은 힌트를 드리자면...

    딱 귀납법으로 풀기 좋은 형태로 보이죠? 먼저 r=3일 때, 즉 점이 3개인 경우를 먼저 풀어봅시다.

    댓글 작성하기 좋아요0 댓글수1
    •  
      베네딕트0724 Lv.6 2020.05.03 09:12

      바로 풀리네요 ㅋㅋㅋㅋ 어이없넹

      좋아요0
  •  
    구머 Lv.6 2020.05.05 09:01

    댓글 작성하기 좋아요0 댓글수0
  •  
    구머 Lv.6 2020.05.16 21:14

    시간이 꽤 지난것 같아 2번 풀이를 공개합니다! 3/4번을 풀 때 도움이 되었으면 합니다

     

    LS_{(1, 0), (p, q)}가 \left \{ \sqrt{x^2+qy^2} : x,y \in \mathbb{N} +\left \{ 0 \right \}\right \}가 됨은 쉽게 관찰할 수 있다. 이제 이 집합의 임의의 두 원소의 곱이 집합의 원소임을 보이면 된다.

     

    pf)

    (a^2 + b^2 q) \times (c^2+d^2 q)=(ac+bdq)^2 +(ac-bd)^2q 이므로 성립함.

    댓글 작성하기 좋아요0 댓글수2
    •  
      파스칼 Lv.7 2020.05.18 07:25

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      상관없긴 한데, LS_{(1,0),(p,q)}=\left \{ \sqrt{x^2+q^2y^2}:x,y\in \mathbb{N}+\left \{ 0 \right \} \right \} 아닌가요?

      그 뒤에 풀이에서도 q => q^2으로 바뀌어야 할 것 같습니다.

      좋아요0
    •  
      구머 Lv.6 2020.05.19 03:58

      수식이 이미지처리가 되어버려서 수정이 원활하지가 않네요. 파스칼님 지적이 맞습니다!

      좋아요0
  •  
    △π Lv.9 2020.05.27 16:08

    ? 저만 invalid eqaution 보이나요?

    댓글 작성하기 좋아요0 댓글수1
    •  
      † J.H † Lv.9 2020.05.28 19:39

      저도 그렇게 보이네요...

      좋아요0
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