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).

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).

Evaluate the following sum (with proof):
\[
\sum_{k=0}^{\infty} \frac{1}{(6k+1)(6k+2)(6k+3)(6k+4)(6k+5)(6k+6)}
\]

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

Here is the best solution of problem 2024-14.

Other solutions were submitted by 권오관 (연세대학교 수학과 22학번, +3), 김준홍 (KAIST 수리과학과 석박통합과정, +3), 김찬우 (연세대학교 수학과 22학번, +3), 노희윤 (KAIST 수리과학과 석박통합과정, +3), 양준혁 (KAIST 수리과학과 20학번, +3), 이명규 (KAIST 전산학부 20학번, +3), 정영훈 (KAIST 새내기과정학부 24학번, +3).

For fixed positive numbers \( x_1, x_2, \dots, x_m \), we define a sequence \( \{ a_n \} \) by \( a_n = x_n \) for \(n \leq m \) and
\[
a_n = a_{n-1}^r + a_{n-2}^r + \dots + a_{n-k}^r
\]
for \( n > m \), where \( r \in (0, 1) \). Find \( \lim_{n \to \infty} a_n \).

The best solution was submitted by 채지석 (KAIST 수리과학과 석박통합과정 21학번, +4). Congratulations!

Here is the best solution of problem 2024-07.

Other solutions were submitted by 김준홍 (KAIST 수리과학과 20학번, +3), 박지운 (KAIST 새내기과정학부 24학번, +3), 정영훈 (KAIST 새내기과정학부 24학번, +3), Anar Rzayev (KAIST 전산학부 19학번, +2), Sasa Sa (+3).

Suppose that we roll \(n\) (6-sided, fair) dice. Let \(S_n\) be the sum of their faces. Find all positive integers \(k\) such that the probability that \(k\) divides \(S_n\) is \(1/k\) for all \(n \geq 1\).

The best solution was submitted by 채지석 (KAIST 수리과학과 석박통합과정 21학번, +4). Congratulations!

Here is the best solution of problem 2024-01.

Other solutions were submitted by 김지원 (KAIST 새내기과정학부 24학번, +3), 김준홍 (KAIST 수리과학과 20학번, +3), 나승균 (KAIST 23학번, +3), 노희윤 (KAIST 수리과학과 석박통합과정 24학번, +4), 신정연 (KAIST 수리과학과 21학번, +3), 신주홍 (KAIST, +3), 심세훈 (KAIST 수리과학과 16학번, +3), 오하빈 (KAIST 수리과학과 19학번, +3), 이명규 (KAIST 전산학부 20학번, +2), 정영훈 (KAIST 새내기과정학부 24학번, +3), 황제민 (KAIST 20학번, +3), 김민서 (KAIST 수리과학과 19학번, +2), 김찬우 (연세대학교 수학과 22학번, +2), 박기윤 (KAIST 수리과학과 23학번, +2). There were incorrect solutions submitted. Late solutions are not graded.

Find all integers \( n \) such that \( n^4 + n^3 + n^2 + n + 1 \) is a perfect square.

The best solution was submitted by 채지석 (KAIST 수리과학과 석박사통학과정 21학번, +4). Congratulations!

Here is the best solution of problem 2023-04.

Other solutions were submitted by 기영인 (KAIST 수리과학과 22학번, +3), 김찬우 (연세대학교 수학과 22학번, +3), 박기윤 (KAIST 새내기과정학부 23학번, +3), 김준홍 (KAIST 수리과학과 20학번, +3), 노희윤 (KAIST 수리과학과 19학번, +3), 문강연 (KAIST 수리과학과 22학번, +3), 이명규 (KAIST 전산학과 20학번, +3), 박지환 (연세대학교 수학과 22학번, +3), 백민수 (원주중학교 교사, +3), 이종서 (KAIST 전산학부 19학번, +3), Matthew Seok, 박준성 (KAIST 수리과학과 석박통합과정 22학번, +3), Anar Rzayev (KAIST 전산학부 19학번, +3), 이동하 (KAIST 새내기과정학부 23학번, +3).

There are light bulbs \(\ell_1,\dots, \ell_n\) controlled by the switches \(s_1, \dots, s_n\). The \(i\)th switch flips the status of the \(i\)th light and possibly others as well. If \(s_i\) flips the status of \(\ell_j\), then \(s_j\) flips the status of \(\ell_i\). All lights are initially off. Prove that it is possible to turn all the lights on.

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

Here is the best solution of problem 2022-24.

Other solutions were submitted by 김기수 (KAIST 수리과학과 18학번, +3), 박준성 (KAIST 수리과학과 석박통합과정, +3).

Define a sequence \( a_n \) by \( a_1 = 1 \) and
\[
a_{n+1} = \frac{1}{n} \left( 1 + \sum_{k=1}^n a_k^2 \right)
\]
for any \( n \geq 1 \). Prove or disprove that \( a_n \) is an integer for all \( n \geq 1 \).

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

Here is the best solution of problem 2022-22.

Other solutions were submitted by 기영인 (KAIST 22학번, +3), 김기수 (KAIST 수리과학과 18학번, +3), 박준성 (KAIST 수리과학과 석박통합과정, +3). An incomplete solution was submitted.

Let \( A \) be an \( n \times n \) Hermitian matrix and \( \lambda_1 (A) \geq \lambda_2 (A) \geq \dots \geq \lambda_n (A) \) the eigenvalues of \( A \). Prove that for any \( 1 \leq k \leq n \)
\[
A \mapsto \lambda_1 (A) + \lambda_2 (A) + \dots + \lambda_k (A)
\]
is a convex function.

The best solution was submitted by 채지석 (수리과학과 2016학번). Congratulations!

Here is his solution of problem 2020-16.

Other solutions were submitted by 길현준 (수리과학과 2018학번, +3), 이준호 (수리과학과 2016학번, +3).

Say there are n points. For each pair of points, we add an edge with probability 1/3. Let \(P_n\) be the probability of the resulting graph to be connected (meaning any two vertices can be joined by an edge path). What can you say about the limit of \(P_n\) as n tends to infinity?

The best solution was submitted by 채지석 (수리과학과 2016학번). Congratulations!

Here is his solution of problem 2020-14.

Other solutions were submitted by 강한필 (전산학부 2016학번, +3), 김건우 (수리과학과 2017학번, +3), 이준호 (수리과학과 2016학번, +3), 김유일 (2020학번, +3).