Prove that for every positive integer \( k \) there exists a positive integer \( n \) such that
\[
\frac{(n+1)(n+2) \dots (2n-k)}{n(n-1) \dots (n-k+1)}
\]
is an integer and that \( k = o(n) \) for such \( n \).

The best solution was submitted by 김은성 (대구과학고등학교, +4). Congratulations!

Here is the best solution of problem 2025-19.

Another solution was submitted by 정영훈 (수리과학과 24학번, +3).

Find all integers \( k \) such that the sequence \( (3n^2 + 3nk^2 + k^3 )_{n=1, 2, \dots} \) contains infinitely many squares.

The best solution was submitted by 김은성 (대구과학고등학교, +4). Congratulations!

Here is the best solution of problem 2025-18.

Other (partial) solutions were submitted by 정영훈 (수리과학과 24학번, +2), Huseyn Ismayilov (전산학부 22학번, +2).