# 2024-05 Knotennullstelle

A complex number $$z \in S^1 \smallsetminus \{1\}$$ is called a Knotennullstelle if there exists a Laurent polynomial $$p(t) \in \mathbb{Z} [t,t^{-1}]$$ such that $$p(1) =\pm 1$$ and $$p(z)=0$$. Prove that the collection of all Knotennullstelle numbers is a discrete subset of $$\mathbb{C}$$.

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# Solution: 2024-04 Real random variable

Prove the following: There exists a bounded real random variable $$Z$$ such that
$E[Z] = 0, E[Z^2] = 1, E[Z^3] = x, E[Z^4] = y$
if and only if $$y \geq x^2 + 1$$. (Here, $$E$$ denotes the expectation.)

The best solution was submitted by 김준홍 (KAIST 수리과학과 20학번, +4). Congratulations!

Other solutions were submitted by 신정연 (KAIST 수리과학과 21학번, +3), 채지석 (KAIST 수리과학과 석박통합과정 21학번, +3), 김지원 (KAIST 새내기과정학부 24학번, +2), 김찬우 (연세대학교 수학과 22학번, +2), 박상현 (고려대학교 수학과 20학번, +2), 이명규 (KAIST 전산학부 20학번, +2), 정영훈 (KAIST 새내기과정학부 24학번, +2). There were incorrect solutions submitted.

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# Solution: 2024-03 Roots of complex derivative

Let $$P(z) = z^3 + c_1 z^2 + c_2 z+ c_3$$ be a complex polynomial in $$\mathbb{C}$$. Its complex derivative is given by $$P’(z) = 3z^{2} +2c_1z+c_{2}.$$ Assume that there exist two points a, b in the open unit disc of complex plane such that P(a) = P(b) =0. Show that  there is a point w belonging to the line segment joining a and b such that  $${\rm Re} (P’(w)) = 0$$.

The best solution was submitted by 김준홍 (KAIST 수리과학과 20학번, +4). Congratulations!

Other solutions were submitted by 김찬우 (연세대학교 수학과 22학번, +3), 노희윤 (KAIST 수리과학과 석박통합과정 24학번, +3), 신정연 (KAIST 수리과학과 21학번, +3), 정영훈 (KAIST 새내기과정학부 24학번, +3), 지은성 (KAIST 수리과학과 23학번, +3), 채지석 (KAIST 수리과학과 석박통합과정 21학번, +3), Anar Rzayev (KAIST 전산학부 19학번, +3), 김지원 (KAIST 새내기과정학부 24학번, +2), 박기윤 (KAIST 수리과학과 23학번, +2), 이명규 (KAIST 전산학부 20학번, +2), There were incorrect solutions submitted.

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# 2024-04 Real random variable

Prove the following: There exists a bounded real random variable $$Z$$ such that
$E[Z] = 0, E[Z^2] = 1, E[Z^3] = x, E[Z^4] = y$
if and only if $$y \geq x^2 + 1$$. (Here, $$E$$ denotes the expectation.)

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# Solution: 2024-02 Well-mixed permutations

A permutation $$\phi \colon \{ 1,2, \ldots, n \} \to \{ 1,2, \ldots, n \}$$ is called a well-mixed if $$\phi (\{1,2, \ldots, k \}) \neq \{1,2, \ldots, k \}$$ for each $$k<n$$. What is the number of well-mixed permutations of $$\{ 1,2, \ldots, 15 \}$$?

The best solution was submitted by 김찬우 (연세대학교 수학과 22학번, +4). Congratulations!

Other solutions were submitted by 김민서 (KAIST 수리과학과 19학번, +3), 김준홍 (KAIST 수리과학과 20학번, +3), 김지원 (KAIST 새내기과정학부 24학번, +3), 노희윤 (KAIST 수리과학과 석박통합과정 24학번, +4), 박기윤 (KAIST 수리과학과 23학번, +3), 신정연 (KAIST 수리과학과 21학번, +3), 이명규 (KAIST 전산학부 20학번, +2), 정영훈 (KAIST 새내기과정학부 24학번, +3), 지은성 (KAIST 수리과학과 23학번, +3). 채지석 (KAIST 수리과학과 석박통합과정 21학번, +3), Sadik Adnan (KAIST 새내기과정학부 23학번, +3), Anar Rzayev (KAIST 전산학부 19학번, +2). There were incorrect solutions submitted.

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Suppose that we roll $$n$$ (6-sided, fair) dice. Let $$S_n$$ be the sum of their faces. Find all positive integers $$k$$ such that the probability that $$k$$ divides $$S_n$$ is $$1/k$$ for all $$n \geq 1$$.