# Solution: 2020-03 Graceful permutations

A permutation $$\pi : [n]\rightarrow [n]$$ is graceful if $$|\pi(i+1) – \pi(i)| \neq |\pi(j+1)-\pi(j)|$$ for all $$i\neq j \in [n-1]$$. For a graceful permutation $$\pi :[2k+1] \rightarrow [2k+1]$$ with $$\pi(\{2,4,\dots,2k\}) = [k]$$, prove that $$\pi(1)+ \pi(2k+1) = 3k+2$$.

The best solution was submitted by 유찬진 (수리과학과 2015학번). Congratulations!

Here is his solution of problem 2020-03.

Other solutions were submitted by 고성훈 (수리과학과 2018학번, +3), 김기수 (수리과학과 2018학번, +3), 김기택 (수리과학과 2015학번, +3), 박현영 (전기및전자공학부 2016학번, +3), 이준호 (2016학번, +3), 조태혁 (수리과학과 2014학번, +3), 채지석 (수리과학과 2016학번, +3), 최고수 (전남과학고등학교, +3).

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# Solution: 2017-18 Limit

Suppose that $$f$$ is differentiable and $\lim_{x\to\infty} (f(x)+f'(x))=2.$  What is $$\lim_{x\to\infty} f(x)$$?

The best solution was submitted by You, Chanjin (유찬진, 수리과학과 2015학번). Congratulations!

Here is his solution of problem 2017-18.

Alternative solutions were submitted by 국윤범 (수리과학과 2015학번, +3), 김태균 (수리과학과 2016학번, +3), 민찬홍 (중앙대학교사범대학부속고등학교 3학년, +3), 이본우 (2017학번, +3), 장기정 (수리과학과 2014학번, +3, alternative solution), 채지석 (2016학번, +3), 최대범 (수리과학과 2016학번, +3), 하석민 (2017학번, +3), Huy Tung Nguyen (수리과학과 2016학번, +3), 윤준기 (전기및전자공학부 2014학번, +2). One incorrect solution was received.

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# Solution: 2017-02 Low-degree polynomial

Let $$a_1,a_2,\ldots,a_n$$ be distinct points in $$\mathbb R^4$$. Does there exist a non-zero polynomial $$P(x_1,x_2,x_3,x_4)$$ such that
(1) the degree of $$P$$ is at most $$\lceil\sqrt{5} n^{1/4}\rceil$$ and
(2) $$P(a_i)=0$$ for all $$i=1,2,\ldots,n$$?

The best solution was submitted by You, Chanjin (유찬진, 수리과학과 2015학번). Congratulations!

Here is his solution of problem 2017-02.

Alternative solutions were submitted by 김태균 (수리과학과 2016학번, +3), 박지민 (전산학부 박사 2017학번, +3), 배형진 (마포고 3학년, +3), 송교범 (고려대 수학과 2017학번, +3), 오동우 (수리과학과 2015학번, +3), 위성군 (수리과학과 2015학번, +3), 이본우 (2017학번, +3), 이시우 (포항공대 수학과 2013학번, +3), 이준호 (2016학번, +3), 장기정 (수리과학과 2014학번, +3), 조태혁 (수리과학과 2014학번, +3), 최대범 (수리과학과 2016학번, +3), 최인혁 (물리학과 2015학번, +3), 홍혁표 (수리과학과 2013학번, +3), Huy Tung Nguyen (2016학번, +3).

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