Let \( X \) and \( Y \) be closed manifolds, and suppose \( X \) is a cover of \( Y \).

 Prove or disprove that the induced map on the first homology is injective.

The best solution was submitted by 신민규 (수리과학과 24학번, +4). Congratulations!

Here is the best solution of problem 2025-07.

Other solutions were submitted by 김동훈 (수리과학과 22학번, +3), Anar Rzayev (수리과학과 19학번, +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).