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).

Prove the following: There exists a bounded real random variable \( Z \) such that
\[
E[Z] = 0, E[Z^2] = 1, E[Z^3] = x, E[Z^4] = y
\]
if and only if \( y \geq x^2 + 1 \). (Here, \( E \) denotes the expectation.)

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

Here is the best solution of problem 2024-04.

Other solutions were submitted by 신정연 (KAIST 수리과학과 21학번, +3), 채지석 (KAIST 수리과학과 석박통합과정 21학번, +3), 김지원 (KAIST 새내기과정학부 24학번, +2), 김찬우 (연세대학교 수학과 22학번, +2), 박상현 (고려대학교 수학과 20학번, +2), 이명규 (KAIST 전산학부 20학번, +2), 정영훈 (KAIST 새내기과정학부 24학번, +2). There were incorrect solutions submitted.

Let \(P(z) = z^3 + c_1 z^2 + c_2 z+ c_3\) be a complex polynomial in \(\mathbb{C}\). Its complex derivative is given by \(P’(z) = 3z^{2} +2c_1z+c_{2}.\) Assume that there exist two points a, b in the open unit disc of complex plane such that P(a) = P(b) =0. Show that  there is a point w belonging to the line segment joining a and b such that  \({\rm Re} (P’(w)) = 0\).

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

Here is the best solution of problem 2024-03.

Other solutions were submitted by 김찬우 (연세대학교 수학과 22학번, +3), 노희윤 (KAIST 수리과학과 석박통합과정 24학번, +3), 신정연 (KAIST 수리과학과 21학번, +3), 정영훈 (KAIST 새내기과정학부 24학번, +3), 지은성 (KAIST 수리과학과 23학번, +3), 채지석 (KAIST 수리과학과 석박통합과정 21학번, +3), Anar Rzayev (KAIST 전산학부 19학번, +3), 김지원 (KAIST 새내기과정학부 24학번, +2), 박기윤 (KAIST 수리과학과 23학번, +2), 이명규 (KAIST 전산학부 20학번, +2), There were incorrect solutions submitted.

A permutation \(\phi \colon \{ 1,2, \ldots, n \} \to \{ 1,2, \ldots, n \}\) is called a well-mixed if \(\phi (\{1,2, \ldots, k \}) \neq \{1,2, \ldots, k \}\) for each \(k<n\). What is the number of well-mixed permutations of \(\{ 1,2, \ldots, 15 \}\)?

The best solution was submitted by 김찬우 (연세대학교 수학과 22학번, +4). Congratulations!

Here is the best solution of problem 2024-02.

Other solutions were submitted by 김민서 (KAIST 수리과학과 19학번, +3), 김준홍 (KAIST 수리과학과 20학번, +3), 김지원 (KAIST 새내기과정학부 24학번, +3), 노희윤 (KAIST 수리과학과 석박통합과정 24학번, +4), 박기윤 (KAIST 수리과학과 23학번, +3), 신정연 (KAIST 수리과학과 21학번, +3), 이명규 (KAIST 전산학부 20학번, +2), 정영훈 (KAIST 새내기과정학부 24학번, +3), 지은성 (KAIST 수리과학과 23학번, +3). 채지석 (KAIST 수리과학과 석박통합과정 21학번, +3), Sadik Adnan (KAIST 새내기과정학부 23학번, +3), Anar Rzayev (KAIST 전산학부 19학번, +2). There were incorrect solutions submitted.

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.

Consider a function \(f: \{1,2,\dots, n\}\rightarrow \mathbb{R}\) satisfying the following for all \(1\leq a,b,c \leq n-2\) with \(a+b+c\leq n\).

\[ f(a+b)+f(a+c)+f(b+c) – f(a)-f(b)-f(c)-f(a+b+c) \geq 0 \text{ and } f(1)=f(n)=0.\]

Prove or disprove this: all such functions \(f\) always have only nonnegative values on its domain.

Acknowledgement: This problem arises during a research discussion between June Huh, Jaehoon Kim and Matt Larson.

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

Here is the best solution of problem 2023-23.

Other solutions were submitted by 김기수 (KAIST 수리과학과 18학번, +3), 김찬우 (연세대학교 수학과 22학번, +3), 박기윤 (KAIST 새내기과정학부 23학번, +3), 지은성 (KAIST 수리과학과 20학번, +3), 이도현 (KAIST 수리과학과 석박통합과정 23학번, +3), 전해구 (KAIST 기계공학과 졸업생, +3).

Does there exist a nontrivial subgroup \(G\) of \( GL(10, \mathbb{C}) \) such that each element in \(G\) is diagonalizable but the set of all the elements of \(G\) is not simultaneously diagonalizable?

The best solution was submitted by 김찬우 (연세대학교 수학과 22학번, +4). Congratulations!

Here is the best solution of problem 2023-22.

Other solutions were submitted by 김기수 (KAIST 수리과학과 18학번, +3), 박기윤 (KAIST 새내기과정학부 23학번, +3), 지은성 (KAIST 수리과학과 20학번, +3), 채지석 (KAIST 수리과학과 석박통합과정 21학번, +3), 이명규 (KAIST 전산학부 20학번, +2).

Find the following limit:

\[
\lim_{n \to \infty} \left( \frac{\sum_{k=1}^{n+2} k^k}{\sum_{k=1}^{n+1} k^k} – \frac{\sum_{k=1}^{n+1} k^k}{\sum_{k=1}^{n} k^k} \right)
\]

The best solution was submitted by 문강연 (KAIST 수리과학과 22학번, +4). Congratulations!

Here is the best solution of problem 2023-21.

Other solutions were submitted by 김기수 (KAIST 수리과학과 18학번, +3), 김준홍 (KAIST 수리과학과 20학번, +3), 김찬우 (연세대학교 수학과 22학번, +3), 박기윤 (KAIST 새내기과정학부 23학번, +3), 이도현 (KAIST 수리과학과 석박통합과정 23학번, +3), 이명규 (KAIST 전산학부 20학번, +3), 지은성 (KAIST 수리과학과 20학번, +3), 채지석 (KAIST 수리과학과 석박통합과정 21학번, +3), Adnan Sadik (KAIST 새내기과정학부 23학번, +3), Muhammadfiruz Hasanov (+3), 조현준 (KAIST 수리과학과 22학번, +2), 서성욱 (대전동산고 2학년, +2).

Can we find a sequence \(a_i, i=0,1,2,…\) with the following property: for each given integer \(n\geq 0\), we have \[\lim_{L\to +\infty}\sum_{i=0}^L 2^{ni} |a_i|\leq 23^{(n+11)^{10}} \quad \text{ and }\quad \lim_{L\to +\infty}\sum_{i=0}^L 2^{ni} a_i = (-1)^n ?\]

The best solution was submitted by 김기수 (KAIST 수리과학과 18학번, +4). Congratulations!

Here is the best solution of problem 2023-20.

Another solution was submitted by 조현준 (KAIST 수리과학과 22학번, +2).