Let \( X \) and \( Y \) be closed manifolds, and suppose \( X \) is a finite-sheeted cover of \( Y \). Prove or disprove that if \( Y \) has a nontrivial first Betti number, then \( X \) also has a nontrivial first Betti number.
The best solution was submitted by Anar Rzayev (수리과학과 19학번, +4). Congratulations!
Here is the best solution of problem 2025-02.
Other solutions were submitted by 김동훈 (수리과학과 22학번, +3), 신민규 (수리과학과 24학번, +3), 성석희 (수리과학과 19학번, +3).
Find all positive integers \( a, n\) such that
\[
\frac{1}{a} + \frac{1}{a+1} + \dots + \frac{1}{a+n}
\]
is an integer.
The best solution was submitted by 박기윤 (전산학부 23학번, +4). Congratulations!
Here is the best solution of problem 2025-01.
Other solutions were submitted by 공기목 (전산학부 20학번, +3), 김동훈 (수리과학과 22학번, +3), 김민서 (수리과학과 19학번, +3), 김준홍 (수리과학과 석박통합과정, +3), 김찬우 (연세대학교 수학과 22학번, +3), 나승균 (수리과학과 23학번, +3), 노희윤 (수리과학과 석박통합과정, +3), 서성욱 (서울대학교 수리과학부 25학번, +3), 신민규 (수리과학과 24학번, +3), 이도엽 (연세대학교 수학과 24학번, +3), 이명규 (전기및전자공학부 20학번, +3), 양준혁 (수리과학과 20학번, +3), 정서윤 (수리과학과 학사과정, +3), 정영훈 (수리과학과 24학번, +3), 채지석 (수리과학과 석박통합과정, +3), 최정담(디지털인문사회과학부 석사과정, +3), 최기범 (한양대학교 졸업생, +3), 최백규 (생명과학과 박사과정, +3). 정지혁 (수리과학과 22학번, +). There were incorrect solutions submitted. Late solutions were not graded.
(Added: The previous best solution has a gap, so we changed the best solution. I apologize for any inconvenience.)
Prove or disprove that every homomorphism \( \pi_1(X) \to \pi_1(X)\) can be realized as the induced homomorphism of a continuous map \(X \to X\).
The best solution was submitted by 김준홍 (KAIST 수리과학과 석박통합과정, +4). Congratulations!
Here is the best solution of problem 2024-21.
Other solutions were submitted by 김찬우 (연세대학교 수학과 22학번, +3), 양준혁 (KAIST 수리과학과 20학번, +3).
Suppose that \( f: \mathbb{R} \to \mathbb{R} \) is a continuous function such that the sequence \( f(x), f(2x), f(3x), \dots \) converges to \( 0 \) for any \( x > 0 \). Prove or disprove that \[ \lim_{x \to \infty} f(x) = 0. \]
The best solution was submitted by 이명규 (KAIST 전산학부 20학번, +4). Congratulations!
Here is the best solution of problem 2024-20.
Other solutions were submitted by 김준홍 (KAIST 수리과학과 석박통합과정, +3), 김찬우 (연세대학교 수학과 22학번, +3), 노희윤 (KAIST 수리과학과 석박통합과정, +3), 양준혁 (KAIST 수리과학과 20학번, +3), 최정담 (KAIST 디지털인문사회과학부 석사과정, +3). There was an incorrect soultion submitted.
Let \(g(t): [0,+\infty) \to [0,+\infty)\) be a decreasing continuous function. Assume \(g(0)=1\), and for every \(s, t \geq 0 \) \[t^{11}g(s+t) \leq 2024 \; [g(s)]^2.\] Show that \(g(11) = g(12)\).
The best solution was submitted by 김준홍 (KAIST 수리과학과 석박통합과정, +4). Congratulations!
Here is the best solution of problem 2024-19.
Other solutions were submitted by 김찬우 (연세대학교 수학과 22학번, +3), 양준혁 (KAIST 수리과학과 20학번, +3), 이명규 (KAIST 전산학부 20학번, +3).
Let \( f(n) \) denote the number of possible sequences of length \( n \), where each term is either \(0, 1,\) or \(-1\), such that the product of every three consecutive numbers is nonnegative. Compute \( f(33)\).
The best solution was submitted by 신민규 (KAIST 새내기과정학부 24학번, +4). Congratulations!
Here is the best solution of problem 2024-18.
Other solutions were submitted by 김준홍 (KAIST 수리과학과 석박통합과정, +3), 김찬우 (연세대학교 수학과 22학번, +3), 노희윤 (KAIST 수리과학과 석박통합과정, +3), 양준혁 (KAIST 수리과학과 20학번, +3), 우준서 (KAIST 수리과학과 20학번, +3), 이명규 (KAIST 전산학부 20학번, +3), 채지석 (KAIST 수리과학과 석박통합과정, +3), 최정담 (KAIST 디지털인문사회과학부 석사과정, +3), Daulet Kurmantayev (+3).
Suppose that \( p(x) \) is a degree \( n \) polynomial with complex coefficients such that \( p(x) \geq 0 \) for any real number \( x \). Prove that
\[
p(x) + p'(x) + \dots + p^{(n)}(x) \geq 0
\]
for any real number \( x \).
The best solution was submitted by 이명규 (KAIST 전산학부 20학번, +4). Congratulations!
Here is the best solution of problem 2024-17.
Other solutions were submitted by 김준홍 (KAIST 수리과학과 석박통합과정, +3), 김찬우 (연세대학교 수학과 22학번, +3), 노희윤 (KAIST 수리과학과 석박통합과정, +3), 서성욱 (대전 동산고 3학년, +3), 양준혁 (KAIST 수리과학과 20학번, +3), 최정담 (KAIST 디지털인문사회과학부 석사과정, +3), 최현준 (KAIST 수리과학과 18학번, +3).
Let \(A= [a_{ij}]_{1\leq i,j\leq 5}\) be a \(5\times 5\) positive definite (real) matrix. Show that the matrix \([a_{ij}/(i+j)]\) is also positive definite.
The best solution was submitted by 김찬우 (연세대학교 수학과 22학번, +4). Congratulations!
Here is the best solution of problem 2024-16.
Other solutions were submitted by 김준홍 (KAIST 수리과학과 석박통합과정, +3), 노희윤 (KAIST 수리과학과 석박통합과정, +3), 서성욱 (대전 동산고 3학년, +3), 신민규 (KAIST 새내기과정학부 24학번, +3), 양준혁 (KAIST 수리과학과 20학번, +3), 이명규 (KAIST 전산학부 20학번, +3), 최정담 (KAIST 디지털인문사회과학부 석사과정, +3).
Is it possible to arrange the numbers \(1, 2, 3, \ldots, 2024\) in a sequence such that the difference between any two adjacent numbers is greater than \(1\) but less than \(4\)?
The best solution was submitted by 김준홍 (KAIST 수리과학과 석박통합과정, +4). Congratulations!
Here is the best solution of problem 2024-15.
Other solutions were submitted by 권오관 (연세대학교 수학과 22학번, +3), 김찬우 (연세대학교 수학과 22학번, +3), 노희윤 (KAIST 수리과학과 석박통합과정, +3), 서성욱 (대전 동산고 3학년, +3), 신민규 (KAIST 새내기과정학부 24학번, +3), 양준혁 (KAIST 수리과학과 20학번, +3), 이명규 (KAIST 전산학부 20학번, +3), 정영훈 (KAIST 새내기과정학부 24학번, +3), 채지석 (KAIST 수리과학과 석박통합과정, +3), 최백규 (KAIST 생명과학과 박사과정, +3), 최정담 (KAIST 디지털인문사회과학부 석사과정, +3), Anar Rzayev (KAIST 전산학부 19학번, +3), ASKM Sayeef Uddin (KAIST 수리과학과 22학번, +3).
Evaluate the following sum (with proof):
\[
\sum_{k=0}^{\infty} \frac{1}{(6k+1)(6k+2)(6k+3)(6k+4)(6k+5)(6k+6)}
\]
The best solution was submitted by 채지석 (KAIST 수리과학과 석박통합과정, +4). Congratulations!
Here is the best solution of problem 2024-14.
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), 양준혁 (KAIST 수리과학과 20학번, +3), 이명규 (KAIST 전산학부 20학번, +3), 정영훈 (KAIST 새내기과정학부 24학번, +3).