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).

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).

Is it possible to arrange the numbers $1, 2, 3, \ldots, 2024$ in a sequence such that the difference between any two adjacent numbers is greater than $1$ but less than $4$?

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

Here is the best solution of problem 2024-15.

Other solutions were submitted by 권오관 (연세대학교 수학과 22학번, +3), 김찬우 (연세대학교 수학과 22학번, +3), 노희윤 (KAIST 수리과학과 석박통합과정, +3), 서성욱 (대전 동산고 3학년, +3), 신민규 (KAIST 새내기과정학부 24학번, +3), 양준혁 (KAIST 수리과학과 20학번, +3), 이명규 (KAIST 전산학부 20학번, +3), 정영훈 (KAIST 새내기과정학부 24학번, +3), 채지석 (KAIST 수리과학과 석박통합과정, +3), 최백규 (KAIST 생명과학과 박사과정, +3), 최정담 (KAIST 디지털인문사회과학부 석사과정, +3), Anar Rzayev (KAIST 전산학부 19학번, +3), ASKM Sayeef Uddin (KAIST 수리과학과 22학번, +3).

Find
$$\sup \left[ \sum_{n=1}^{\infty} \frac{1}{\sqrt{n}} \left( \sum_{i=n}^{\infty} x_i^2 \right)^{1/2} \Big/ \sum_{i=1}^{\infty} x_i \right],$$
where the supremum is taken over all monotone decreasing sequences of positive numbers $(x_i)$ such that $\sum_{i=1}^{\infty} x_i < \infty$.

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

Here is the best solution of problem 2024-10.

There were incorrect solutions submitted.

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.

Let $f(t)=(t^{pq}-1)(t-1)$ and $g(t)=(t^{p}-1)(t^q-1)$ where $p$ and $q$ are relatively prime positive integers. Prove that $\frac{f(t)}{g(t)}$ can be written as a polynomial where it has just $1$ or $-1$ as coefficients. (For example, when $p=2$ and $q=3$, we have that $\frac{f(t)}{g(t)} = t^2-t+1$.)

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

Here is the best solution of problem 2023-14.

Other solutions were submitted by 강지민 (세마고 3학년, +3), 김기수 (KAIST 수리과학과 18학번, +3), 김민서 (KAIST 수리과학과 19학번, +3), 김찬우 (연세대학교 수학과 22학번, +3), 이도현 (KAIST 수리과학과 석박통합과정 23학번, +3), 이명규 (KAIST 전산학부 20학번, +3), 전해구 (KAIST 기계공학과 졸업생, +3), 조현준 (KAIST 수리과학과 22학번, +4), 지은성 (KAIST 수리과학과 20학번, +3), 채지석 (KAIST 수리과학과 석박통합과정 21학번, +3), Anar Rzayev (KAIST 전산학부 19학번, +3). Muhammadfiruz Hasanov (+3).

Let $\{x_1, x_2, \ldots, x_{21}\} = \{-10, -9, \ldots, -1, 0, 1, \ldots, 9, 10\}$. What is the largest possible value of $x_1x_2x_3+x_4x_5x_6+\cdots + x_{19}x_{20}x_{21}$?

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

Here is the best solution of problem 2023-05.