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.

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 \(n\geq 1.\)

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

Here is the best solution of problem 2024-09.

Other solutions were submitted by 김준홍 (KAIST 수리과학과 20학번, +3), 김지원 (KAIST 새내기과정학부 24학번, +3), 박지운 (KAIST 새내기과정학부 24학번, +3), 신정연 (KAIST 수리과학과 21학번, +3), 이명규 (KAIST 전산학부 20학번, +3), 정영훈 (KAIST 새내기과정학부 24학번, +3), 채지석 (KAIST 수리과학과 석박통합과정 21학번, +3), Anar Rzayev (KAIST 전산학부 19학번, +3).