Trefoil to Figure-Eight by Crossing Changes

A knot is a simple closed curve in three-dimensional space. A diagram of a knot is a projection of the knot to the plane together with over/under information at each crossing. A crossing change is the operation of switching one crossing from over to under or from under to over.

Find up to three examples of sequences of knot diagrams that begin with a diagram of the trefoil knot and end with a diagram of the figure-eight knot, where each step is obtained from the previous diagram by a crossing change.

The obvious chains trefoil → unknot → figure-eight and trefoil → trefoil # figure-eight → figure-eight are not allowed.

(3 points will be given for three correct examples, 2 points for two correct examples, and 1 point for one correct example.)

The best solution was submitted by 이원준 (Rutgers University, +4). Congratulations!

Here is the best solution of problem 2026-08.

Other solutions were submitted by 신민규 (수리과학과 24학번, +3), 정서윤 (수리과학과 23학번, +3), 장현준 (서울과학고 3학년, +2).

Let
\[
Q_n=\{0,1\}^n
\]
be the \(n\)-dimensional discrete cube, viewed as a graph in which two vertices are adjacent if they differ in exactly one coordinate.

For a subset \(A\subseteq Q_n\), let \(\partial_e A\) denote the set of edges with one endpoint in \(A\) and the other in \(Q_n\setminus A\).

Prove that for every \(A\subseteq Q_n\),
\[
|\partial_e A|\ge |A|\bigl(n-\log_2|A|\bigr).
\]

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

Here is the best solution of problem 2026-07.

Other solutions were submitted by 김은성 (서울대 수리과학부, +3), 신민규 (수리과학과 24학번, +3), 장현준 (서울과학고 3학년, +3), 정서윤 (수리과학과 23학번, +3), 최완수 (서울대 물리천문학부, +3).

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).

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).

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).

We want to find the maximum area of two disjoint, simply connected, congruent tiles that can be packed inside a right triangle, one of whose angles is \( \pi/6 \) (30 degrees). What would be the maximal coverage of the right triangle by the tiles? (There is no restriction on the shape of the tiles, especially it does not need to be rectangular, as long as they are simply connected.) (4 points will be given to the one with the best answer, and 3 points for the next four best answers.)

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

Here is the best solution of problem 2026-01.

Another solution was submitted by 김은성 (서울대 수리과학과, +3), 김준홍 (수리과학과 석박통합과정, +3), 김찬우 (연세대학교 수학과, +3), 박영우 (전산학부 24학번, +3), 신민규 (수리과학과 24학번, +3), 이상주 (경남대 수학교육과, +3), 정영훈 (수리과학과 24학번, +3), Huseyn Ismayilov (전산학부 22학번, +3), 정서윤 (수리과학과 23학번, +2).

Prove that for every positive integer \( k \) there exists a positive integer \( n \) such that
\[
\frac{(n+1)(n+2) \dots (2n-k)}{n(n-1) \dots (n-k+1)}
\]
is an integer and that \( k = o(n) \) for such \( n \).

The best solution was submitted by 김은성 (대구과학고등학교, +4). Congratulations!

Here is the best solution of problem 2025-19.

Another solution was submitted by 정영훈 (수리과학과 24학번, +3).

Find all integers \( k \) such that the sequence \( (3n^2 + 3nk^2 + k^3 )_{n=1, 2, \dots} \) contains infinitely many squares.

The best solution was submitted by 김은성 (대구과학고등학교, +4). Congratulations!

Here is the best solution of problem 2025-18.

Other (partial) solutions were submitted by 정영훈 (수리과학과 24학번, +2), Huseyn Ismayilov (전산학부 22학번, +2).