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[대한수학회] 대41. 유리수점? 정수점? 곱가능집합?

수학동아 2020.05.01 09:11 조회 2873

페이스북 트위터 카카오톡 카카오스토리 네이버블로그 네이버밴드

대한수학회 41번

유리수점? 정수점? 곱가능집합?

문제 출제자 : 오병권 서울대학교 수리과학부 교수

좌표평면 상의 점 $(p, q)$ 에 대해 $p, q$ 가 모두 유리수이면 $(p, q)$ 를 유리수점이라 정의하고,

$p, q$ 가 모두 정수이면 $(p, q)$ 를 정수점이라 정의한다.

좌표평면 상의 점 $(p, q), (r, s)$ 와 임의의 실수 $t$ 에 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

$S_{(a_1, b_1), (a_2, b_2), \cdots , (a_r, b_r)}=$ { $x_1(a_1, b_1)+x_2(a_2, b_2)+\cdots +x_r(a_r, b_r) :x_1,\cdots ,x_r$ 은 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 $E$ 가, 모든 $x, y \in E$ 에 대해 $xy \in E$ 를 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 $m$ 과 $3m^2-1$ 을 나누는 소수 $p$ 에 대해, $LS_{(1, m), (0, p)}$ 는 곱가능집합임을 보여라.

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

이전글 대40. 연속함수의 근사

다음글 대42. 도전! 무한 연분수

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해결

도전

구머 Lv.6 2020.05.01 10:51 비밀댓글

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댓글 작성하기 댓글수6
- 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
  
  넵
  
  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

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
도전

유한의끝도못본남자 Lv.7 2020.05.01 21:47 비밀댓글

확인요청중

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- 수학꿀잼 Lv.3 2020.05.07 03:51
  
  혹시 몇번인가요?
  
  좋아요0
도전

파스칼 Lv.7 2020.05.02 07:11 비밀댓글

확인요청중

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Function: parseLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view.php
Line: 343
Function: view

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
Line: 558
Function: view

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

A PHP Error was encountered

Severity: Warning

Message: file_put_contents(/DATA/upload/polymath/latex/1358e3d1002ae0bc6fa177e7ce794732.gif): failed to open stream: No such file or directory

Filename: libraries/Common.php

Line Number: 213

Backtrace:

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 213
Function: file_put_contents

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 236
Function: getLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view_comment_list.php
Line: 90
Function: parseLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view.php
Line: 343
Function: view

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
Line: 558
Function: view

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

A PHP Error was encountered

Severity: Warning

Message: mkdir(): Permission denied

Filename: libraries/Common.php

Line Number: 202

Backtrace:

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 202
Function: mkdir

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 236
Function: getLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view_comment_list.php
Line: 90
Function: parseLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view.php
Line: 343
Function: view

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
Line: 558
Function: view

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

A PHP Error was encountered

Severity: Warning

Message: file_put_contents(/DATA/upload/polymath/latex/cfbf11c5c04e654a3f29669cd8a5fb51.gif): failed to open stream: No such file or directory

Filename: libraries/Common.php

Line Number: 213

Backtrace:

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 213
Function: file_put_contents

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 236
Function: getLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view_comment_list.php
Line: 90
Function: parseLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view.php
Line: 343
Function: view

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
Line: 558
Function: view

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

A PHP Error was encountered

Severity: Warning

Message: mkdir(): Permission denied

Filename: libraries/Common.php

Line Number: 202

Backtrace:

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 202
Function: mkdir

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 236
Function: getLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view_comment_list.php
Line: 90
Function: parseLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view.php
Line: 343
Function: view

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
Line: 558
Function: view

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

A PHP Error was encountered

Severity: Warning

Message: file_put_contents(/DATA/upload/polymath/latex/ee87517f57f2ed1d16b7ade8385a1fd1.gif): failed to open stream: No such file or directory

Filename: libraries/Common.php

Line Number: 213

Backtrace:

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 213
Function: file_put_contents

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 236
Function: getLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view_comment_list.php
Line: 90
Function: parseLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view.php
Line: 343
Function: view

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
Line: 558
Function: view

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

A PHP Error was encountered

Severity: Warning

Message: mkdir(): Permission denied

Filename: libraries/Common.php

Line Number: 202

Backtrace:

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 202
Function: mkdir

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 236
Function: getLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view_comment_list.php
Line: 90
Function: parseLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view.php
Line: 343
Function: view

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
Line: 558
Function: view

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

A PHP Error was encountered

Severity: Warning

Message: file_put_contents(/DATA/upload/polymath/latex/c319e0ac8182c461897c6c6499d85dc9.gif): failed to open stream: No such file or directory

Filename: libraries/Common.php

Line Number: 213

Backtrace:

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 213
Function: file_put_contents

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 236
Function: getLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view_comment_list.php
Line: 90
Function: parseLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view.php
Line: 343
Function: view

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
Line: 558
Function: view

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

A PHP Error was encountered

Severity: Warning

Message: mkdir(): Permission denied

Filename: libraries/Common.php

Line Number: 202

Backtrace:

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 202
Function: mkdir

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 236
Function: getLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view_comment_list.php
Line: 90
Function: parseLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view.php
Line: 343
Function: view

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
Line: 558
Function: view

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

A PHP Error was encountered

Severity: Warning

Message: file_put_contents(/DATA/upload/polymath/latex/2922bdaab0516525130b3bd95e8a9a7f.gif): failed to open stream: No such file or directory

Filename: libraries/Common.php

Line Number: 213

Backtrace:

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 213
Function: file_put_contents

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 236
Function: getLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view_comment_list.php
Line: 90
Function: parseLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view.php
Line: 343
Function: view

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
Line: 558
Function: view

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

A PHP Error was encountered

Severity: Warning

Message: mkdir(): Permission denied

Filename: libraries/Common.php

Line Number: 202

Backtrace:

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 202
Function: mkdir

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 236
Function: getLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view_comment_list.php
Line: 90
Function: parseLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view.php
Line: 343
Function: view

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
Line: 558
Function: view

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

A PHP Error was encountered

Severity: Warning

Message: file_put_contents(/DATA/upload/polymath/latex/ee87517f57f2ed1d16b7ade8385a1fd1.gif): failed to open stream: No such file or directory

Filename: libraries/Common.php

Line Number: 213

Backtrace:

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 213
Function: file_put_contents

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 236
Function: getLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view_comment_list.php
Line: 90
Function: parseLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view.php
Line: 343
Function: view

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
Line: 558
Function: view

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

A PHP Error was encountered

Severity: Warning

Message: mkdir(): Permission denied

Filename: libraries/Common.php

Line Number: 202

Backtrace:

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 202
Function: mkdir

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 236
Function: getLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view_comment_list.php
Line: 90
Function: parseLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view.php
Line: 343
Function: view

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
Line: 558
Function: view

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

A PHP Error was encountered

Severity: Warning

Message: file_put_contents(/DATA/upload/polymath/latex/ee87517f57f2ed1d16b7ade8385a1fd1.gif): failed to open stream: No such file or directory

Filename: libraries/Common.php

Line Number: 213

Backtrace:

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 213
Function: file_put_contents

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 236
Function: getLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view_comment_list.php
Line: 90
Function: parseLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view.php
Line: 343
Function: view

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
Line: 558
Function: view

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

A PHP Error was encountered

Severity: Warning

Message: mkdir(): Permission denied

Filename: libraries/Common.php

Line Number: 202

Backtrace:

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 202
Function: mkdir

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 236
Function: getLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view_comment_list.php
Line: 90
Function: parseLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view.php
Line: 343
Function: view

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
Line: 558
Function: view

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

A PHP Error was encountered

Severity: Warning

Message: file_put_contents(/DATA/upload/polymath/latex/ee87517f57f2ed1d16b7ade8385a1fd1.gif): failed to open stream: No such file or directory

Filename: libraries/Common.php

Line Number: 213

Backtrace:

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 213
Function: file_put_contents

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 236
Function: getLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view_comment_list.php
Line: 90
Function: parseLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view.php
Line: 343
Function: view

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
Line: 558
Function: view

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

A PHP Error was encountered

Severity: Warning

Message: mkdir(): Permission denied

Filename: libraries/Common.php

Line Number: 202

Backtrace:

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 202
Function: mkdir

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 236
Function: getLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view_comment_list.php
Line: 90
Function: parseLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view.php
Line: 343
Function: view

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
Line: 558
Function: view

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

A PHP Error was encountered

Severity: Warning

Message: file_put_contents(/DATA/upload/polymath/latex/2922bdaab0516525130b3bd95e8a9a7f.gif): failed to open stream: No such file or directory

Filename: libraries/Common.php

Line Number: 213

Backtrace:

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 213
Function: file_put_contents

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 236
Function: getLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view_comment_list.php
Line: 90
Function: parseLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view.php
Line: 343
Function: view

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
Line: 558
Function: view

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

A PHP Error was encountered

Severity: Warning

Message: mkdir(): Permission denied

Filename: libraries/Common.php

Line Number: 202

Backtrace:

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 202
Function: mkdir

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 236
Function: getLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view_comment_list.php
Line: 90
Function: parseLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view.php
Line: 343
Function: view

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
Line: 558
Function: view

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

A PHP Error was encountered

Severity: Warning

Message: file_put_contents(/DATA/upload/polymath/latex/ee87517f57f2ed1d16b7ade8385a1fd1.gif): failed to open stream: No such file or directory

Filename: libraries/Common.php

Line Number: 213

Backtrace:

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 213
Function: file_put_contents

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 236
Function: getLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view_comment_list.php
Line: 90
Function: parseLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view.php
Line: 343
Function: view

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
Line: 558
Function: view

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

A PHP Error was encountered

Severity: Warning

Message: mkdir(): Permission denied

Filename: libraries/Common.php

Line Number: 202

Backtrace:

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 202
Function: mkdir

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 236
Function: getLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view_comment_list.php
Line: 90
Function: parseLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view.php
Line: 343
Function: view

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
Line: 558
Function: view

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

A PHP Error was encountered

Severity: Warning

Message: file_put_contents(/DATA/upload/polymath/latex/cfbf11c5c04e654a3f29669cd8a5fb51.gif): failed to open stream: No such file or directory

Filename: libraries/Common.php

Line Number: 213

Backtrace:

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 213
Function: file_put_contents

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 236
Function: getLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view_comment_list.php
Line: 90
Function: parseLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view.php
Line: 343
Function: view

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
Line: 558
Function: view

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

A PHP Error was encountered

Severity: Warning

Message: mkdir(): Permission denied

Filename: libraries/Common.php

Line Number: 202

Backtrace:

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 202
Function: mkdir

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 236
Function: getLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view_comment_list.php
Line: 90
Function: parseLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view.php
Line: 343
Function: view

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
Line: 558
Function: view

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

A PHP Error was encountered

Severity: Warning

Message: file_put_contents(/DATA/upload/polymath/latex/0f87b1d9d288a0a027fd3153d70b4000.gif): failed to open stream: No such file or directory

Filename: libraries/Common.php

Line Number: 213

Backtrace:

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 213
Function: file_put_contents

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 236
Function: getLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view_comment_list.php
Line: 90
Function: parseLatexImg

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view.php
Line: 343
Function: view

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
Line: 558
Function: view

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

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- 파스칼 Lv.7 2020.05.02 08:29
  
  1번 풀이입니다.
  
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- 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!
  
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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.

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- 베네딕트0724 Lv.6 2020.05.03 09:12
  
  바로 풀리네요 ㅋㅋㅋㅋ 어이없넹
  
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구머 Lv.6 2020.05.05 09:01

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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.

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- 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.
  
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- 구머 Lv.6 2020.05.19 03:58
  
  수식이 이미지처리가 되어버려서 수정이 원활하지가 않네요. 파스칼님 지적이 맞습니다!
  
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△π Lv.9 2020.05.27 16:08

? 저만 invalid eqaution 보이나요?

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- † J.H † Lv.9 2020.05.28 19:39
  
  저도 그렇게 보이네요...
  
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