For given \(a, b \in \mathbb{R}\) and \(c \in \mathbb{Z}\), find all function \(f: \mathbb{R} \to \mathbb{R}\) which is continuous at 0 and satisfies
\[
f(ax) = f(bx) + x^c \quad \forall x\in \mathbb{R}\setminus \{0\}.
\]
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).