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 \(\mathbb{E} \left[ \max_i \frac{|X_i|}{\sqrt{1 + \log i}} \right] \leq C\).
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).
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 \(\lim_{n\rightarrow \infty} \frac{\mathbb E[X]}{n^2}\).
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).
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
\[
f(ax) = f(bx) + x^c \quad \forall x\in \mathbb{R}\setminus \{0\}.
\]
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).
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.
There are \(n+1\) hats, each labeled with a number from \(1\) to \(n+1\), and \(n\) people. Each person is randomly assigned exactly one hat, and each hat is assigned to at most one person (i.e., the assignment is injective). A person can see all other assigned hats but cannot see their own hat and the unassigned hat. Each person must independently guess the number on their own hat.
If everyone correctly guesses their own hat’s number, they win; otherwise, they lose. They may discuss a strategy before the hats are assigned, but no communication is allowed afterward.
Determine a strategy that maximizes their probability of winning.
The best solution was submitted by 김동훈 (수리과학과 22학번, +4). Congratulations!
Here is the best solution of problem 2025-06.
Other solutions were submitted by 김준홍 (수리과학과 석박통합과정, +3), 박기윤 (전산학부 23학번, +3), 채지석 (수리과학과 석박통합과정, +3). There were incorrect solutions submitted.
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.
There are \(n+1\) hats, each labeled with a number from \(1\) to \(n+1\), and \(n\) people. Each person is randomly assigned exactly one hat, and each hat is assigned to at most one person (i.e., the assignment is injective). A person can see all other assigned hats but cannot see their own hat and the unassigned hat. Each person must independently guess the number on their own hat.
If everyone correctly guesses their own hat’s number, they win; otherwise, they lose. They may discuss a strategy before the hats are assigned, but no communication is allowed afterward.
Determine a strategy that maximizes their probability of winning.