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[슬기로운 수학생활] 슬18. 함.풀.문 cttg친구 문제의 활용

수학동아 2021.10.01 01:01 조회 1349

페이스북 트위터 카카오톡 카카오스토리 네이버블로그 네이버밴드

슬기로운 수학생활 18번

함.풀.문 cttg친구 문제의 활용

문제 출제자 : 백진언 미시간 대학교 수학과 박사과정생

1번. 임의의 정수 $a, b$ 에 대해, $a-b|f(a)-f(b)$ 를 만족하는 다항식이 아닌 함수 $f: \mathbb{Z}\rightarrow \mathbb{Z}$ 가 존재하는가?

위 문제는 '세상을 바꿀 문제'에서 cttg 친구의 문제입니다.

http://www.polymath.co.kr/contents/view/32482?page=1

2번. 만약 1번에 대한 답이 '존재한다'라면, $f$ 에 어떤 조건을 추가해야 $f$ 가 다항식일 수 밖에 없을까?

2-1번. 어떤 양수 $C$ , $e$ 가 있어, $f$ 가 1번의 성질을 만족하고, 모든 정수 $n$ 에 대해 $f(n)\leq C\left | n \right |^{e}$ 도 만족한다고 하자. 그렇다면 $f$ 는 다항식인가?

2-2번 이외에도 1번의 조건에 추가하면 $f$ 가 무조건 다항식이 될 수 있는 성질들을 찾아보고, 정말 그런지 증명하거나 반증해보자

이전글 슬17. 그래프의 삼각형

다음글 슬19. 정사각형 안에서 돌아가는 도형

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수현인데닉넴누가먹었 Lv.6 2021.10.03 03:07

흠....

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일반성을 잃고 Lv.5 2021.11.14 07:19 비밀댓글

비밀 댓글이 등록 되었습니다.

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- 출제자(슬기) Lv.4 2022.01.12 19:28 비밀댓글
  
  비밀 댓글이 등록 되었습니다!
일반성을 잃고 Lv.5 2021.11.14 07:29

2-2번은... 왠지 코시함수방정식 (f(x+y)=f(x)+f(y))의 실수확장조건이랑 관련있을 것 같습니다.

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ln(-1)=πi Lv.3 2021.12.16 03:57

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$a-b | f(a)-f(b)$ 는 $a-b$ 가 $f(a)-f(b)$ 의 약수라는 의미인가요?

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- 조영준 멘토 Lv.3 2022.01.12 02:26
  
  네! a-b가 f(a)-f(b)를 나눈다는 의미입니다~
  
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진리를 찾아서 Lv.5 2022.02.15 20:30

1번 풀이

다항식인 함수 f(x)에서 x를 [x]로 바꾼 모든 함수는 조건을 만족합니다.

답 : 존재한다

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- 출제자(슬기) Lv.4 2022.03.24 15:32
  
  함수 f의 정의역은 실수가 아니라 정수입니다. 그래서 floor function을 적용해도 함수 값이 바뀌지 않아요.
  
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- 진리를 찾아서 Lv.5 2022.03.24 21:12
  
  아..너무 쉽게 생각했나 봅니다. 다시 고민해보겠습니다.
  
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