Let \(p\) be a prime number at least three and let \(k\) be a positive integer smaller than \(p\). Given \(a_1,\dots, a_k\in \mathbb{F}_p\) and distinct elements \(b_1,\dots, b_k\in \mathbb{F}_p\), prove that there exists a permutation \(\sigma\) of \([k]\) such that the values of \(a_i + b_{\sigma(i)}\) are distinct modulo \(p\).

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

Here is the best solution of problem 2023-12.

Other solutions were submitted by 김찬우 (연세대학교 수학과 22학번, +3), 박준성 (KAIST 수리과학과 석박통합과정 22학번, +3), 채지석 (KAIST 수리과학과 석박통합과정 21학번, +3), 최민규 (한양대학교 의학대학 졸업, +3), Anar Rzayev (KAIST 전산학부 19학번, +3). Late solutions were not graded.

Let \(S\) be a set of distinct \(20\) integers. A set \(T_A\) is defined as \(T_A:=\{ s_1+s_2+s_3 \mid s_1, s_2, s_3 \in S\}\). What is the smallest possible cardinality of \(T_A\)?

The best solution was submitted by 정희승 (서울대학교 물리천문학부, +4). Congratulations!

Here is the best solution of problem 2023-11.

Other solutions were submitted by 김찬우 (연세대학교 수학과 22학번, +3), 박기윤 (새내기과정학부 23학번, +3), 박준성 (KAIST 수리과학과 석박통합과정 22학번, +3), 신민서(KAIST 수리과학과 20학번, +3), 채지석 (KAIST 수리과학과 석박통합과정 21학번, +3), 최민규 (한양대학교 의학대학 졸업, +3), Eun Song (+3), James Hamilton Clerk (+3), Anar Rzayev (KAIST 전산학부 19학번, +2), 김준홍 (KAIST 수리과학과 20학번, +2), 최백규 (KAIST 수리과학과 석박통합과정 21학번, +3).

Find all pairs of prime numbers \( (p, q) \) such that \( pq \) divides \( p^p + q^q + 1 \).

The best solution was submitted by 최백규 (KAIST 생명과학과 박사과정 20학번, +4). Congratulations!

Here is the best solution of problem 2023-10.

Other solutions were submitted by 김준홍 (KAIST 수리과학과 20학번, +3), 김찬우 (연세대학교 수학과 22학번, +3), 박준성 (KAIST 수리과학과 석박통합과정 22학번, +3), 신민서(KAIST 수리과학과 20학번, +3), 전해구 (KAIST 기계공학과 졸업, +3), 채지석 (KAIST 수리과학과 석박통합과정 21학번, +3), Anar Rzayev (KAIST 전산학부 19학번, +3), Matthew Seok (+3).

Let \(\mathbb{S}_n\) be the set of all permutations of \([n]=\{1,\dots, n\}\). For positive real numbers \(d_1,\dots, d_n\), prove \[ \sum_{\sigma\in \mathbb{S}_n} \frac{1}{ d_{\sigma(1)}(d_{\sigma(1)}+d_{\sigma(2)}) \dots (d_{\sigma(1)}+\dots + d_{\sigma(n)}) } = \frac{1}{d_1\dots d_n}.\]

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

Here is the best solution of problem 2023-09.

Other solutions were submitted by 권도현 (KAIST 수리과학과 22학번, +3), 김명규 (KAIST 전산학부 19학번, +3), 김준홍 (KAIST 수리과학과 20학번, +3), 김찬우 (연세대학교 수학과 22학번, +3), 박기윤 (KAIST 새내기과정학부 23학번, +3), 박준성 (KAIST 수리과학과 석박통합과정 22학번, +3),이명규 (KAIST 전산학부 20학번, +3), 채지석 (KAIST 수리과학과 석박통합과정 21학번, +3), Anar Rzayev (KAIST 전산학부 19학번, +3). James Hamilton Clerk (+3), Matthew Seok (+3).

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).

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.