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 \( \sum_{i=1}^{\infty} x_i < \infty \).
Let \(A\) be a \(16 \times 16\) matrix whose entries are either \(1\) or \(-1\). What is the maximum value of the determinant of \(A\)?
The best solution was submitted by 이명규 (KAIST 전산학부 20학번, +4).
Congratulations!
Here is the best solution of problem 2024-08.
Other solutions were submitted by 김준홍 (KAIST 수리과학과 20학번, +3), 김지원 (KAIST 새내기과정학부 24학번, +3), 신정연 (KAIST 수리과학과 21학번, +3), 정영훈 (KAIST 새내기과정학부 24학번, +3), 지은성 (KAIST 수리과학과 20학번, +3), 채지석 (KAIST 수리과학과 석박통합과정 21학번, +3), Anar Rzayev (KAIST 전산학부 19학번, +3), 권오관 (연세대학교 수학과 22학번, +2).
Find all positive numbers \(a_1,…,a_{5}\) such that \(a_1^\frac{1}{n} + \cdots + a_{5}^\frac{1}{n}\) is integer for every integer \(n\geq 1.\)
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).
It is found that there is a flaw in POW 2024-05; some students showed that the collection of all Knotennullstelle numbers is not a discrete subset of \( \mathbb{C} \). We again apologize for the inconvenience.
To acknowledge the students who reported the flaws in POW 2024-05 and POW 2024-06, we decided to give credits to 김준홍 (KAIST 수리과학과 20학번, +4) and 지은성 (KAIST 수리과학과 20학번, +3) for POW 2024-05 and Anar Rzayev (KAIST 전산학부 19학번, +4) for POW 2024-06.
Here is a “solution” of problem 2024-05.
Let \(A\) be a \(16 \times 16\) matrix whose entries are either \(1\) or \(-1\). What is the maximum value of the determinant of \(A\)?
It is found that there is a flaw in POW 2024-06; the inequality in the problem is not satisfied with the given g(t). Since it is too late to revise the problem again with a new deadline, we decide to cancel POW 2024-06. We apologize for the inconvenience.
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 \( \lim_{n \to \infty} a_n \).
Due to the change of the assumption in POW 2024-06, the due date for the submitting the solution is postponed to May 13 (Mon.) 3PM. (Originally, it was 3PM Friday.)
In POW 2024-06, there was a typo in the assumption. It is now corrected that the assumption holds for \( t \in [-1, 1] \). (Originally, it was for \( t \in \mathbb{R} \).)