Find a pair of nonisomorphic nonabelian groups so that their abelianizations are isomorphic and their commutator subgroups are perfect.
The best solution was submitted by 김찬우 (연세대학교 수학과 22학번, +4). Congratulations!
Here is the best solution of problem 2023-08.
Other solutions were submitted by 박기윤 (KAIST 새내기과정학부 23학번, +3), 이명규 (KAIST 전산학과 20학번, +3), Anar Rzayev (KAIST 전산학부 19학번, +2).
Suppose that \( f: [a, b] \to \mathbb{R} \) is a smooth, convex function, and there exists a constant \( t>0 \) such that \( f'(x) \geq t \) for all \( x \in (a, b) \). Prove that
\[
\left| \int_a^b e^{i f(x)} dx \right| \leq \frac{2}{t}.
\]
The best solution was submitted by Anar Rzayev (KAIST 전산학부 19학번, +4). Congratulations!
Here is the best solution of problem 2023-07.
Other solutions were submitted by 김찬우 (연세대학교 수학과 22학번, +3), 박기윤 (KAIST 새내기과정학부 23학번, +3), 박준성 (KAIST 수리과학과 석박통합과정 22학번, +3), 오현섭 (KAIST 수리과학과 박사과정 21학번, +3), 이명규 (KAIST 전산학과 20학번, +3), 최예준 (서울과학기술대학교 행정학과 21학번, +3), Matthew Seok (+3), James Hamilton Clerk (+3).
Let \(\phi = \frac{1+\sqrt{5}}{2}\). Let \(f(1)=1\) and for \(n\geq 1\), let
\[ f(n+1) = \left\{\begin{array}{ll}
f(n)+2 & \text{ if } f(f(n)-n+1)=n \\
f(n)+1 & \text{ otherwise}.
\end{array}\right.\]
Prove that \(f(n) = \lfloor \phi n \rfloor\), and determine when \(f(f(n)-n+1)\neq n\) holds.
The best solution was submitted by 박기윤 (KAIST 새내기과정학부 23학번, +4). Congratulations!
Here is the best solution of problem 2023-06.
Other solutions were submitted by 김찬우 (연세대학교 수학과 22학번, +3), 이동하 (KAIST 새내기과정학부 23학번, +3), 최예준 (서울과학기술대학교 행정학과 21학번, +3), Matthew Seok (+2). Late solutions are not graded.
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.
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).
Determine the minimum number of hyperplanes in \(\mathbb{R}^n\) that do not contain the origin but they together cover all points in \(\{0,1\}^n\) except the origin.
The best solution was submitted by 이종서 (KAIST 전산학부 19학번, +4). Congratulations!
Here is the best solution of problem 2023-03.
Other solutions were submitted by 김찬우 (연세대학교 수학과 22학번, +3), 박기윤 (KAIST 새내기과정학부 23학번, +3), 박준성 (KAIST 수리과학과 석박통합과정 22학번, +3). There were two incorrect solutions submitted.
Suppose \( a_1, a_2, \dots, a_{2023} \) are real numbers such that
\[
a_1^3 + a_2^3 + \dots + a_n^3 = (a_1 + a_2 + \dots + a_n)^2
\]
for any \( n = 1, 2, \dots, 2023 \). Prove or disprove that \( a_n \) is an integer for any \( n = 1, 2, \dots, 2023 \).
The best solution was submitted by 기영인 (KAIST 수리과학과 22학번, +4). Congratulations!
Here is the best solution of problem 2023-01.
Other solutions were submitted by 고성훈 (KAIST 수리과학과 18학번, +3), 김찬우 (연세대학교 수학과 22학번, +3), 박기윤 (KAIST 새내기과정학부 23학번, +3), 임도현 (KAIST 수리과학과 22학번, +3), 신정여 (KAIST 수리과학과 21학번, +3), 문강연 (KAIST 수리과학과 22학번, +3), 이명규 (KAIST 전산학과 20학번, +3), 박현영 (KAIST 전기및전자공학부 석박사통합과정 22학번, +3), Myint Mo Zwe (KAIST 새내기과정학부 22학번, +3), 이재경 (KAIST 뇌인지과학과 22학번, +3), Matthew Seok, 김기수 (KAIST 수리과학과 18학번, +3), 박준성 (KAIST 수리과학과 석박통합과정 22학번, +3), Yusuf Bahadir Kilicarslan (KAIST 전산학부 19학번, +3), 이동하 (KAIST 새내기과정학부 23학번, +2). Late solutions are not graded.
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).
Let \(A\) be an 8 by 8 integral unimodular matrix. Moreover, assume that for each \( x \in \mathbb{Z}^8 \), we have \(x^{\top} A x \) is even. What is the possible number of positive eigenvalues for \(A\)?
The best solution was submitted by Noitnetta Yobepyh (Snaejwen High School, +4). Congratulations!
Here is the best solution of problem 2022-23.
Other solutions were submitted by 김기수 (KAIST 수리과학과 18학번, +3), 여인영 (KAIST 물리학과 20학번, +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.