Solution: 2025-11 Maxima of standard Gaussian

Let \( X_1, X_2, \ldots \) be an infinite sequence of standard normal random variables which are not necessarily independent. Show that there exists a universal constant \( C \) such that \(\mathbb{E} \left[ \max_i \frac{|X_i|}{\sqrt{1 + \log i}} \right] \leq C\).

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

Here is the best solution of problem 2025-11.

Other solutions were submitted by 김동훈 (수리과학과 22학번, +3), 정서윤 (수리과학과 학사과정, +3), Anar Rzayev (수리과학과 19학번, +3).

Solution: 2025-10 Intersections of random chords

Let \(P\) be a regular \(2n\)-gon. A perfect matching is a partition of vertex points into \(n\) unordered pairs; each pair represents a chord drawn inside \(P\). Two chords are said to “intersect” if they have a nonempty intersection.

Let \(X\) be the (random) number of intersection points (formed by intersecting chords) in a perfect matching chosen uniformly at random from the set of all possible matchings. Note that more than two chords can intersect at the same point, and in this case this intersection point is just counted once. Compute \(\lim_{n\rightarrow \infty} \frac{\mathbb E[X]}{n^2}\).

The best solution was submitted by Anar Rzayev (수리과학과 19학번, +4). Congratulations!

Here is the best solution of problem 2025-10.

Other solutions were submitted by 김동훈 (수리과학과 22학번, +3), 김준홍 (수리과학과 석박통합과정, +3), 신민규 (수리과학과 24학번, +3), 정서윤 (수리과학과 학사과정, +2).

2025-11 Maxima of standard Gaussian

Let \( X_1, X_2, \ldots \) be an infinite sequence of standard normal random variables which are not necessarily independent. Show that there exists a universal constant \( C \) such that \(\mathbb{E} \left[ \max_i \frac{|X_i|}{\sqrt{1 + \log i}} \right] \leq C\).

Solution: 2025-09 abc-functions

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).

2025-10 Intersections of random chords

Let \(P\) be a regular \(2n\)-gon. A perfect matching is a partition of vertex points into \(n\) unordered pairs; each pair represents a chord drawn inside \(P\). Two chords are said to “intersect” if they have a nonempty intersection.

Let \(X\) be the (random) number of intersection points (formed by intersecting chords) in a perfect matching chosen uniformly at random from the set of all possible matchings. Note that more than two chords can intersect at the same point, and in this case this intersection point is just counted once. Compute \(\lim_{n\rightarrow \infty} \frac{\mathbb E[X]}{n^2}\).

Solution: 2025-08 Chordial relations

Consider a convex \((n+2)\)-gon. Let \(a_n\)​ denote the number of ways to add non-crossing chords to this polygon, including the case where no chords are added (i.e., \(a_0=a_1=0\) and \(a_2=3\)).

Find a recurrence relation for the sequence \(a_n​\) and determine its generating function.

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

Here is the best solution of problem 2025-08.

Other solutions were submitted by 김동훈 (수리과학과 22학번, +3), 신민규 (수리과학과 24학번, +3), 정서윤 (수리과학과 학사과정, +3), 채지석 (수리과학과 석박통합과정, +3), Anar Rzayev (수리과학과 19학번, +3).

2025-09 abc-functions

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

Solution: 2025-07 Do Covers Induce Injective Maps on Homology

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).

2025-08 Chordial relations

Consider a convex \((n+2)\)-gon. Let \(a_n\)​ denote the number of ways to add non-crossing chords to this polygon, including the case where no chords are added (i.e., \(a_0=a_1=1\) and \(a_2=3\)).

Find a recurrence relation for the sequence \(a_n​\) and determine its generating function.