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.
Let \( X \in \mathbb{R}^{n \times n} \) be a symmetric matrix with eigenvalues \( \lambda_i \) and orthonormal eigenvectors \( u_i \). The spectral decomposition gives \( X = \sum_{i=1}^n \lambda_i u_i u_i^\top \). For a function \( f : \mathbb{R} \to \mathbb{R} \), define \( f(X) := \sum_{i=1}^n f(\lambda_i) u_i u_i^\top \). Let \( X, Y \in \mathbb{R}^{n \times n} \) be symmetric. Is it always true that \( e^{X+Y} = e^X e^Y \)? If not, under what conditions does the equality hold?
The best solution was submitted by 이명규 (전기및전자공학부 20학번, +4). Congratulations!
Here is the best solution of problem 2025-05.
Other solutions were submitted by 김동훈 (수리과학과 22학번, +3), 김준홍 (수리과학과 석박통합과정, +3), 신민규 (수리과학과 24학번, +3), 정서윤 (수리과학과 학사과정, +3), 채지석 (수리과학과 석박통합과정, +3).
Let \( X \in \mathbb{R}^{n \times n} \) be a symmetric matrix with eigenvalues \( \lambda_i \) and orthonormal eigenvectors \( u_i \). The spectral decomposition gives \( X = \sum_{i=1}^n \lambda_i u_i u_i^\top \). For a function \( f : \mathbb{R} \to \mathbb{R} \), define \( f(X) := \sum_{i=1}^n f(\lambda_i) u_i u_i^\top \). Let \( X, Y \in \mathbb{R}^{n \times n} \) be symmetric. Is it always true that \( e^{X+Y} = e^X e^Y \)? If not, under what conditions does the equality hold?
We write \(tx = (tx_0,…,tx_5)\) for \(x=(x_0,…,x_5)\in \mathbb{R^{6}}\) and \(t\in \mathbb{R}\). Find all real multivariate polynomials \(P(x)\) in \(x\) satisfying the following properties:
(a) \(P(tx) = t^d P(x)\) for all \(t\in \mathbb{R}\) and \(x\in \mathbb{R}^{6}\), where \(0\leq d \leq 15\) is an integer;
(b) \(P(x) =0\) if \(x_i = x_j\) with \(i\neq j\).
The best solution was submitted by 채지석 (수리과학과 석박통합과정, +4). Congratulations!
Here is the best solution of problem 2025-04.
Other solutions were submitted by 이명규 (전기및전자공학부 20학번), 김동훈 (수리과학과 22학번, +3), 김준홍 (수리과학과 석박통합과정, +3), 신민규 (수리과학과 24학번, +3), 정서윤 (수리과학과 학사과정, +3), Anar Rzayev (수리과학과 19학번, +3).
Consider any sequence \( a_1,\dots, a_n \) of non-negative integers in \(\{0,1,\dots, m\}\). Prove that \[|\{ a_i+ a_j + (j-i): 1\leq i < j \leq n \}|\geq m \] when \(m= \lfloor \frac{1}{4} n^{2/3} \rfloor \).
The best solution was submitted by 김준홍 (수리과학과 석박통합과정, +4). Congratulations!
Here is the best solution of problem 2025-03.
Another solution was submitted by Anar Rzayev (수리과학과 19학번, +2).
We write \(tx = (tx_0,…,tx_5)\) for \(x=(x_0,…,x_5)\in \mathbb{R^{6}}\) and \(t\in \mathbb{R}\). Find all real multivariate polynomials \(P(x)\) in \(x\) satisfying the following properties:
(a) \(P(tx) = t^d P(x)\) for all \(t\in \mathbb{R}\) and \(x\in \mathbb{R}^{6}\), where \(0\leq d \leq 15\) is an integer;
(b) \(P(x) =0\) if \(x_i = x_j\) with \(i\neq j\).
Let \( X \) and \( Y \) be closed manifolds, and suppose \( X \) is a finite-sheeted cover of \( Y \). Prove or disprove that if \( Y \) has a nontrivial first Betti number, then \( X \) also has a nontrivial first Betti number.
The best solution was submitted by Anar Rzayev (수리과학과 19학번, +4). Congratulations!
Here is the best solution of problem 2025-02.
Other solutions were submitted by 김동훈 (수리과학과 22학번, +3), 신민규 (수리과학과 24학번, +3), 성석희 (수리과학과 19학번, +3).
Consider any sequence \( a_1,\dots, a_n \) of non-negative integers in \(\{0,1,\dots, m\}\). Prove that \[|\{ a_i+ a_j + (j-i): 1\leq i < j \leq n \}|\geq m \] when \(m= \lfloor \frac{1}{4} n^{2/3} \rfloor \).
A bonus problem: Can you find a function \(f(n)=\omega(n^{2/3})\) such that the above statement is true when \(m = f(n) \)? Is there such a function with \(f(n)= \Omega(n)\)? (You would still get full points without answering the bonus question.)
Find all positive integers \( a, n\) such that
\[
\frac{1}{a} + \frac{1}{a+1} + \dots + \frac{1}{a+n}
\]
is an integer.
The best solution was submitted by 박기윤 (전산학부 23학번, +4). Congratulations!
Here is the best solution of problem 2025-01.
Other solutions were submitted by 공기목 (전산학부 20학번, +3), 김동훈 (수리과학과 22학번, +3), 김민서 (수리과학과 19학번, +3), 김준홍 (수리과학과 석박통합과정, +3), 김찬우 (연세대학교 수학과 22학번, +3), 나승균 (수리과학과 23학번, +3), 노희윤 (수리과학과 석박통합과정, +3), 서성욱 (서울대학교 수리과학부 25학번, +3), 신민규 (수리과학과 24학번, +3), 이도엽 (연세대학교 수학과 24학번, +3), 이명규 (전기및전자공학부 20학번, +3), 양준혁 (수리과학과 20학번, +3), 정서윤 (수리과학과 학사과정, +3), 정영훈 (수리과학과 24학번, +3), 채지석 (수리과학과 석박통합과정, +3), 최정담(디지털인문사회과학부 석사과정, +3), 최기범 (한양대학교 졸업생, +3), 최백규 (생명과학과 박사과정, +3). 정지혁 (수리과학과 22학번, +). There were incorrect solutions submitted. Late solutions were not graded.
(Added: The previous best solution has a gap, so we changed the best solution. I apologize for any inconvenience.)