Find all positive integers \( a, n\) such that
\[
\frac{1}{a} + \frac{1}{a+1} + \dots + \frac{1}{a+n}
\]
is an integer.

The best solution was submitted by 박기윤 (전산학부 23학번, +4). Congratulations!

Here is the best solution of problem 2025-01.

Other solutions were submitted by 공기목 (전산학부 20학번, +3), 김동훈 (수리과학과 22학번, +3), 김민서 (수리과학과 19학번, +3), 김준홍 (수리과학과 석박통합과정, +3), 김찬우 (연세대학교 수학과 22학번, +3), 나승균 (수리과학과 23학번, +3), 노희윤 (수리과학과 석박통합과정, +3), 서성욱 (서울대학교 수리과학부 25학번, +3), 신민규 (수리과학과 24학번, +3), 이도엽 (연세대학교 수학과 24학번, +3), 이명규 (전기및전자공학부 20학번, +3), 양준혁 (수리과학과 20학번, +3), 정서윤 (수리과학과 학사과정, +3), 정영훈 (수리과학과 24학번, +3), 채지석 (수리과학과 석박통합과정, +3), 최정담(디지털인문사회과학부 석사과정, +3), 최기범 (한양대학교 졸업생, +3), 최백규 (생명과학과 박사과정, +3). 정지혁 (수리과학과 22학번, +). There were incorrect solutions submitted. Late solutions were not graded.

(Added: The previous best solution has a gap, so we changed the best solution. I apologize for any inconvenience.)

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.