Let \(f(x)\) be a function such that \((1-x^2) f”(x) – 2x f'(x) + \alpha (\alpha+1) f(x) =0\)
for some \(\alpha \not\in \mathbb{N}\). Define \(P_n (x) = \frac{d^n}{dx^n} (x^2-1)^n\) for \(n =0,1,…\). Compute \(\int_{-1}^1 f(x) P_n(x) dx.\)

The best solution was submitted by 정서윤 (수리과학과 23학번, +4). Congratulations!

Here is the best solution of problem 2026-06.

Other solutions were submitted by 김은성 (서울대 수리과학부, +3), 신민규 (수리과학과 24학번, +3), 이상주 (경남대 수학교육과, +3), 장현준 (서울과학고 3학년, +3), 지은성 (수리과학과 석박통합과정, +3), Huseyn Ismayilov (전산학부 22학번, +3).

Find all positive integer \( k \) satisfying the following statement: For any positive integers \( m \) and \( n \),
\[
\frac{((k+1)m)! ((k+1)n)!}{m! n! ((k-1)m + n)! (m + (k-1)n)!}
\]
is an integer.

The best solution was submitted by 정서윤 (수리과학과 23학번, +4). Congratulations!

Here is the best solution of problem 2026-02.

Another solution was submitted by 김은성 (서울대 수리과학과, +3), 박영우 (전산학부 24학번, +3), 신민규 (수리과학과 24학번, +3), 장현준 (서울과학고 3학년, +3), Huseyn Ismayilov (전산학부 22학번, +3).

Prove the following identity:
\[
\sum_{k=0}^{n-1} \binom{z}{k} \frac{x^{n-k}}{n-k} = \sum_{k=1}^n \binom{z-k}{n-k} \frac{(x+1)^k -1}{k}.
\]

The best solution was submitted by 정서윤 (수리과학과 23학번, +4). Congratulations!

Here is the best solution of problem 2025-17.

Other solutions were submitted by 김은성 (대구과학고, +3), 김찬우 (연세대학교 수학과, +3), 정영훈 (수리과학과 24학번, +3), Huseyn Ismayilov (전산학부 22학번, +3).

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
\[
f(ax) = f(bx) + x^c \quad \forall x\in \mathbb{R}\setminus \{0\}.
\]

The best solution was submitted by 정서윤 (수리과학과 학사과정, +4). Congratulations!

Here is the best solution of problem 2025-09.

Other solutions were submitted by 김동훈 (수리과학과 22학번, +3), 신민규 (수리과학과 24학번, +3), 채지석 (수리과학과 석박통합과정, +3), Anar Rzayev (수리과학과 19학번, +2), 김준홍 (수리과학과 석박통합과정, +2), 이명규 (전기및전자공학부 20학번, +2).