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).
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.
Suppose that \( f: [a, b] \to \mathbb{R} \) is a smooth, convex function, and there exists a constant \( t>0 \) such that \( f'(x) \geq t \) for all \( x \in (a, b) \). Prove that
\[
\left| \int_a^b e^{i f(x)} dx \right| \leq \frac{2}{t}.
\]
The best solution was submitted by Anar Rzayev (KAIST 전산학부 19학번, +4). Congratulations!
Here is the best solution of problem 2023-07.
Other solutions were submitted by 김찬우 (연세대학교 수학과 22학번, +3), 박기윤 (KAIST 새내기과정학부 23학번, +3), 박준성 (KAIST 수리과학과 석박통합과정 22학번, +3), 오현섭 (KAIST 수리과학과 박사과정 21학번, +3), 이명규 (KAIST 전산학과 20학번, +3), 최예준 (서울과학기술대학교 행정학과 21학번, +3), Matthew Seok (+3), James Hamilton Clerk (+3).