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).

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).

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.)