Let \(n\) be an odd positive integer, and let
\[
f:\{-1,1\}^n\to\{-1,1\}.
\]
Interpret \(x_i=1\) as voter \(i\) voting for candidate \(A\), and \(x_i=-1\) as voter \(i\) voting for candidate \(B\). The value \(f(x_1,\dots,x_n)\) is the choice.

Find all functions \(f\) satisfying the following properties:
1. Anonymity: for every permutation \(\sigma\in S_n\),
\[
f(x_1,\dots,x_n)=f(x_{\sigma(1)},\dots,x_{\sigma(n)}).
\]
2. Neutrality:
\[
f(-x_1,\dots,-x_n)=-f(x_1,\dots,x_n).
\]
3. Monotonicity: if \(x=(x_1,\dots,x_n)\) and \(y=(y_1,\dots,y_n)\) satisfy
\[
x_i\le y_i \qquad \text{for all } i=1,\dots,n,
\]
then
\[
f(x)\le f(y).
\]

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

Here is the best solution of problem 2026-04.

Other solutions were submitted by 기영인 (+3), 김범석 (인하대, +3), 김은성 (서울대 수리과학과, +3), 김준홍 (수리과학과 석박통합과정, +4), 이상주 (경남대 수학교육과, +3), 이재원 (새내기과정학부 26학번, +3), 장현준 (서울과학고 3학년, +3), 정서윤 (수리과학과 23학번, +3), 지은성 (수리과학과 석박통합과정, +3), Huseyn Ismayilov (전산학부 22학번, +3).

Let \(V\) be the set of tuples \((a_1,…,a_5)\) such that \(a_1 \leq a_2 \leq \cdots \leq a_5 \) belong to \(\mathbb{R}\) and satisfy \[ \sum_{1\leq i\leq 5} a_i >0, \quad \sum_{1\leq i<j \leq 5} a_i a_j >0, \quad \sum_{1\leq i<j< k \leq 5} a_ia_ja_k >0.\]

What is the maximum number \(p\) such that there exists a tuple \((a_1,…,a_5) \) in \(V\) whose \(a_p\leq 0 \)?

The best solution was submitted by 김준홍 (수리과학과 석박통합과정, +4). Congratulations!

Here is the best solution of problem 2026-03.

Other solutions were submitted by 김범석 (인하대, +3), 김은성 (서울대 수리과학과, +3), 김찬우 (연세대 수학과, +3), 신민규 (수리과학과 24학번, +3), 이상주 (경남대 수학교육과, +3), 장현준 (서울과학고 3학년, +3), 정서윤 (수리과학과 23학번, +3), 지은성 (수리과학과 석박통합과정, +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).