Let \(V\) be the set of tuples \((a_1,…,a_5)\) such that \(a_1 \leq a_2 \leq \cdots \leq a_5 \) belong to \(\mathbb{R}\) and satisfy \[ \sum_{1\leq i\leq 5} a_i >0, \quad \sum_{1\leq i<j \leq 5} a_i a_j >0, \quad \sum_{1\leq i<j< k \leq 5} a_ia_ja_k >0.\]
What is the maximum number \(p\) such that there exists a tuple \((a_1,…,a_5) \) in \(V\) whose \(a_p\leq 0 \)?
We want to find the maximum area of two disjoint, simply connected, congruent tiles that can be packed inside a right triangle, one of whose angles is \( \pi/6 \) (30 degrees). What would be the maximal coverage of the right triangle by the tiles? (There is no restriction on the shape of the tiles, especially it does not need to be rectangular, as long as they are simply connected.) (4 points will be given to the one with the best answer, and 3 points for the next four best answers.)
The best solution was submitted by 지은성 (수리과학과 석박통합과정, +4). Congratulations!
Here is the best solution of problem 2026-01.
Another solution was submitted by 김은성 (서울대 수리과학과, +3), 김준홍 (수리과학과 석박통합과정, +3), 김찬우 (연세대학교 수학과, +3), 박영우 (전산학부 24학번, +3), 신민규 (수리과학과 24학번, +3), 이상주 (경남대 수학교육과, +3), 정영훈 (수리과학과 24학번, +3), Huseyn Ismayilov (전산학부 22학번, +3), 정서윤 (수리과학과 23학번, +2).
Find all positive integer \( k \) satisfying the following statement: For any positive integers \( m \) and \( n \),
\[
\frac{((k+1)m)! ((k+1)n)!}{m! n! ((k-1)m + n)! (m + (k-1)n)!}
\]
is an integer.
The email account pow@mathsci.kaist.ac.kr was not working normally. It is now fixed.
POW2026-01 is revised to clarify the problem. (The revision is only for the clarification and there is essentially no change in the problem.)
We want to find the maximum area of two disjoint, simply connected, congruent tiles that can be packed inside a right triangle, one of whose angles is \( \pi/6 \) (30 degrees). What would be the maximal coverage of the right triangle by the tiles? (There is no restriction on the shape of the tiles, especially it does not need to be rectangular, as long as they are simply connected.) (4 points will be given to the one with the best answer, and 3 points for the next four best answers.)