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.)

Prove or disprove that every homomorphism \( \pi_1(X) \to \pi_1(X)\) can be realized as the induced homomorphism of a continuous map \(X \to X\).

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

Here is the best solution of problem 2024-21.

Other solutions were submitted by 김찬우 (연세대학교 수학과 22학번, +3), 양준혁 (KAIST 수리과학과 20학번, +3).

Suppose that \( f: \mathbb{R} \to \mathbb{R} \) is a continuous function such that the sequence \( f(x), f(2x), f(3x), \dots \) converges to \( 0 \) for any \( x > 0 \). Prove or disprove that \[ \lim_{x \to \infty} f(x) = 0. \]

The best solution was submitted by 이명규 (KAIST 전산학부 20학번, +4). Congratulations!

Here is the best solution of problem 2024-20.

Other solutions were submitted by 김준홍 (KAIST 수리과학과 석박통합과정, +3), 김찬우 (연세대학교 수학과 22학번, +3), 노희윤 (KAIST 수리과학과 석박통합과정, +3), 양준혁 (KAIST 수리과학과 20학번, +3), 최정담 (KAIST 디지털인문사회과학부 석사과정, +3). There was an incorrect soultion submitted.

Let \(g(t): [0,+\infty) \to [0,+\infty)\) be a decreasing continuous function. Assume \(g(0)=1\), and for every \(s, t \geq 0 \) \[t^{11}g(s+t) \leq 2024 \; [g(s)]^2.\] Show that \(g(11) = g(12)\).

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

Here is the best solution of problem 2024-19.

Other solutions were submitted by 김찬우 (연세대학교 수학과 22학번, +3), 양준혁 (KAIST 수리과학과 20학번, +3), 이명규 (KAIST 전산학부 20학번, +3).

Let \( f(n) \) denote the number of possible sequences of length \( n \), where each term is either \(0, 1,\) or \(-1\), such that the product of every three consecutive numbers is nonnegative. Compute \( f(33)\).

The best solution was submitted by 신민규 (KAIST 새내기과정학부 24학번, +4). Congratulations!

Here is the best solution of problem 2024-18.

Other solutions were submitted by 김준홍 (KAIST 수리과학과 석박통합과정, +3), 김찬우 (연세대학교 수학과 22학번, +3), 노희윤 (KAIST 수리과학과 석박통합과정, +3), 양준혁 (KAIST 수리과학과 20학번, +3), 우준서 (KAIST 수리과학과 20학번, +3), 이명규 (KAIST 전산학부 20학번, +3), 채지석 (KAIST 수리과학과 석박통합과정, +3), 최정담 (KAIST 디지털인문사회과학부 석사과정, +3), Daulet Kurmantayev (+3).

Suppose that \( p(x) \) is a degree \( n \) polynomial with complex coefficients such that \( p(x) \geq 0 \) for any real number \( x \). Prove that
\[
p(x) + p'(x) + \dots + p^{(n)}(x) \geq 0
\]
for any real number \( x \).

The best solution was submitted by 이명규 (KAIST 전산학부 20학번, +4). Congratulations!

Here is the best solution of problem 2024-17.

Other solutions were submitted by 김준홍 (KAIST 수리과학과 석박통합과정, +3), 김찬우 (연세대학교 수학과 22학번, +3), 노희윤 (KAIST 수리과학과 석박통합과정, +3), 서성욱 (대전 동산고 3학년, +3), 양준혁 (KAIST 수리과학과 20학번, +3), 최정담 (KAIST 디지털인문사회과학부 석사과정, +3), 최현준 (KAIST 수리과학과 18학번, +3).