Category Archives: solution

Solution: 2024-11 Rationals from real polynomials

Find all polynomials \( P \) with real coefficients such that \( P(x) \in \mathbb{Q} \) implies \( x \in \mathbb{Q} \).

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

Here is the best solution of problem 2024-11.

Other solutions were submitted by 권오관 (연세대학교 수학과 22학번, +3), 김준홍 (KAIST 수리과학과 석박통합과정, +3), 김찬우 (연세대학교 수학과 22학번, +3), 노희윤 (KAIST 수리과학과 석박통합과정, +3), 이명규 (KAIST 전산학부 20학번, +3), 김지원 (KAIST 새내기과정학부 24학번, +2), 정영훈 (KAIST 새내기과정학부 24학번, +2).

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Solution: 2024-10 Supremum

Find
\[
\sup \left[ \sum_{n=1}^{\infty} \frac{1}{\sqrt{n}} \left( \sum_{i=n}^{\infty} x_i^2 \right)^{1/2} \Big/ \sum_{i=1}^{\infty} x_i \right],
\]
where the supremum is taken over all monotone decreasing sequences of positive numbers \( (x_i) \) such that \( \sum_{i=1}^{\infty} x_i < \infty \).

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

Here is the best solution of problem 2024-10.

There were incorrect solutions submitted.

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Solution: 2024-09 Integer sums

Find all positive numbers \(a_1,…,a_{5}\) such that \(a_1^\frac{1}{n} + \cdots + a_{5}^\frac{1}{n}\) is integer for every integer \(n\geq 1.\)

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

Here is the best solution of problem 2024-09.

Other solutions were submitted by 김준홍 (KAIST 수리과학과 20학번, +3), 김지원 (KAIST 새내기과정학부 24학번, +3), 박지운 (KAIST 새내기과정학부 24학번, +3), 신정연 (KAIST 수리과학과 21학번, +3), 이명규 (KAIST 전산학부 20학번, +3), 정영훈 (KAIST 새내기과정학부 24학번, +3), 채지석 (KAIST 수리과학과 석박통합과정 21학번, +3), Anar Rzayev (KAIST 전산학부 19학번, +3).

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Solution: 2024-08 Determinants of 16 by 16 matricies

Let \(A\) be a \(16 \times 16\) matrix whose entries are either \(1\) or \(-1\). What is the maximum value of the determinant of \(A\)?

The best solution was submitted by 이명규 (KAIST 전산학부 20학번, +4).

Congratulations!

Here is the best solution of problem 2024-08.

Other solutions were submitted by 김준홍 (KAIST 수리과학과 20학번, +3), 김지원 (KAIST 새내기과정학부 24학번, +3), 신정연 (KAIST 수리과학과 21학번, +3), 정영훈 (KAIST 새내기과정학부 24학번, +3), 지은성 (KAIST 수리과학과 20학번, +3), 채지석 (KAIST 수리과학과 석박통합과정 21학번, +3), Anar Rzayev (KAIST 전산학부 19학번, +3), 권오관 (연세대학교 수학과 22학번, +2).

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Solution: 2024-07 Limit of a sequence

For fixed positive numbers \( x_1, x_2, \dots, x_m \), we define a sequence \( \{ a_n \} \) by \( a_n = x_n \) for \(n \leq m \) and
\[
a_n = a_{n-1}^r + a_{n-2}^r + \dots + a_{n-k}^r
\]
for \( n > m \), where \( r \in (0, 1) \). Find \( \lim_{n \to \infty} a_n \).

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

Here is the best solution of problem 2024-07.

Other solutions were submitted by 김준홍 (KAIST 수리과학과 20학번, +3), 박지운 (KAIST 새내기과정학부 24학번, +3), 정영훈 (KAIST 새내기과정학부 24학번, +3), Anar Rzayev (KAIST 전산학부 19학번, +2), Sasa Sa (+3).

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Notice on POW 2024-05 and POW 2024-06

It is found that there is a flaw in POW 2024-05; some students showed that the collection of all Knotennullstelle numbers is not a discrete subset of \( \mathbb{C} \). We again apologize for the inconvenience.

To acknowledge the students who reported the flaws in POW 2024-05 and POW 2024-06, we decided to give credits to 김준홍 (KAIST 수리과학과 20학번, +4) and 지은성 (KAIST 수리과학과 20학번, +3) for POW 2024-05 and Anar Rzayev (KAIST 전산학부 19학번, +4) for POW 2024-06.

Here is a “solution” of problem 2024-05.

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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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