A link in S³ is a smooth embedding of a finite disjoint union of circles into S³. A link diagram is a generic projection to S² together with over/under data at each double point. For an oriented 2-component link K ∪ J, the linking number lk(K, J) is one-half of the signed sum of the crossings between K and J.

Prove or disprove that if lk(K, J) = 0, then there exist disjoint, compact, properly embedded, orientable surfaces F₁, F₂ ⊂ S³ × I such that

∂F₁ = K × {1}
∂F₂ = J × {1}.

Your solution should consist almost entirely of pictures. Each picture may have at most one short explanatory sentence.

(It turns out that the converse is also true.)

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

Here is the best solution of problem 2026-05.

Other solutions were submitted by 장현준 (서울과학고 3학년, +3), 정서윤 (수리과학과 23학번, +2).

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).

Each punch can be centered anywhere in the plane and removes all points within distance 1 from its center. What is the minimum number of punches needed to remove every point in the annulus between the circles of radius 7 and 10 (with the same center)? Describe your construction. The person with the smallest number of punches earns +4, and the next four best answers earn +3.

The best solutions were submitted by 신민규 (수리과학과 24학번, +4) and 김준홍 (수리과학과 석박통합과정, +4). Congratulations!

Here are the best solutions of problem 2025-13 (solution 1, solution 2).

Other solutions were submitted by 김찬우 (연세대 수학과, +3), 정서윤 (수리과학과 학사과정, +3), 지은성 (수리과학과 석박통합과정, +3), 정영훈 (수리과학과 24학번, +2), 김은성 (대구과학고, +3), 김지원 (전산학부 24학번, +2), Anar Rzayev (수리과학과 19학번, +2).

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).