2024-06 Limit of concave functions

Let \(f_n(t)\), \(n=1,2…\) be a sequence of concave functions on \(\mathbb{R}\). Assume \(\liminf_{n\to\infty} f_n(t) \geq 2024\,t^{5}+3\) for \(t\in [-1, 1]\) and \(\lim_{n\to \infty} f_n(0) = 3\). Suppose \(f_n'(0)\) exist for \(n=1,2,…\). Compute \(\lim_{n\to \infty} f_n'(0)\).

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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!

Here is the best solution of problem 2024-04.

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!

Here is the best solution of problem 2024-03.

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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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!

Here is the best solution of problem 2024-02.

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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Solution: 2024-01 Dice

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 \(n \geq 1\).

The best solution was submitted by 채지석 (KAIST 수리과학과 석박통합과정 21학번, +4). Congratulations!

Here is the best solution of problem 2024-01.

Other solutions were submitted by 김지원 (KAIST 새내기과정학부 24학번, +3), 김준홍 (KAIST 수리과학과 20학번, +3), 나승균 (KAIST 23학번, +3), 노희윤 (KAIST 수리과학과 석박통합과정 24학번, +4), 신정연 (KAIST 수리과학과 21학번, +3), 신주홍 (KAIST, +3), 심세훈 (KAIST 수리과학과 16학번, +3), 오하빈 (KAIST 수리과학과 19학번, +3), 이명규 (KAIST 전산학부 20학번, +2), 정영훈 (KAIST 새내기과정학부 24학번, +3), 황제민 (KAIST 20학번, +3), 김민서 (KAIST 수리과학과 19학번, +2), 김찬우 (연세대학교 수학과 22학번, +2), 박기윤 (KAIST 수리과학과 23학번, +2). There were incorrect solutions submitted. Late solutions are not graded.

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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  \({\rm Re} (P’(w)) = 0\).

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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 \(\{ 1,2, \ldots, 15 \}\)?

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