Show that any set of d + 2 points in R^d can be partitioned into two sets whose convex hulls intersect.
The best solution was submitted by 정영훈 (수리과학과 24학번, +4). Congratulations!
Here is the best solution of problem 2025-14.
Other solutions were submitted by 김은성 (대구과학고, +3), 김지원 (전산학부 24학번, +3), 김찬우 (연세대 수학과, +3), 신민규 (수리과학과 24학번, +3), 이태민 (경남대 수학교육과, +3), 정서윤 (수리과학과 학사과정, +3), 지은성 (수리과학과 석박통합과정, +3).
Each punch can be centered anywhere in the plane and removes all points within distance 1 from its center. What is the minimum number of punches needed to remove every point in the annulus between the circles of radius 7 and 10 (with the same center)? Describe your construction. The person with the smallest number of punches earns +4, and the next four best answers earn +3.
The best solutions were submitted by 신민규 (수리과학과 24학번, +4) and 김준홍 (수리과학과 석박통합과정, +4). Congratulations!
Here are the best solutions of problem 2025-13 (solution 1, solution 2).
Other solutions were submitted by 김찬우 (연세대 수학과, +3), 정서윤 (수리과학과 학사과정, +3), 지은성 (수리과학과 석박통합과정, +3), 정영훈 (수리과학과 24학번, +2), 김은성 (대구과학고, +3), 김지원 (전산학부 24학번, +2), Anar Rzayev (수리과학과 19학번, +2).
Currently, the email address pow@mathsci.kaist.ac.kr is not working normally. Please submit your solutions to jioon.lee@kaist.edu instead. The due (for both POW 2025-13 and POW 2025-14) is postponed to Oct. 3 (Fri.) 23:59 pm.
Since there were no solutions submitted for POW 2025-13, we will postpone the due date of POW 2025-13 to Oct. 3 (Fri.) 3PM.
We remark that the problem is designed so that the answer with smallest number of punches earns +4, and the next four best answers earn +3 each. (In other words, you may earn some points even if your answer is not optimal.)
Show that any set of d + 2 points in R^d can be partitioned into two sets whose convex hulls intersect.
Find all positive integers \( a, b \) such that
\[
\frac{1}{a} + \frac{1}{b} = \frac{p_1}{p_2}
\]
where \( p_1 \) and \( p_2 \) are consecutive primes.
The best solution was submitted by 지은성 (수리과학과 석박통합과정, +4). Congratulations!
Here is the best solution of problem 2025-12.
Other solutions were submitted by 김동훈 (수리과학과 22학번, +3), 김민찬 (수리과학과 22학번, +3), 김은성 (대구과학고, +3), 김준홍 (수리과학과 석박통합과정, +3), 김찬우 (연세대 수학과, +3), 노희윤 (수리과학과 석박통합과정, +3), +3), 신민규 (수리과학과 24학번, +3), 우준서 (전기및전자공학부 20학번, +3), 이상주 (경남대 수학교육과, +3), 정서윤 (수리과학과 학사과정, +3), 정영훈 (수리과학과 24학번, +3), 채지석 (수리과학과 석박통합과정, +3), Anar Rzayev (수리과학과 19학번, +3), Muhammad Hasnat Farooq (+3), 최백규 (생명과학과 박사과정, +2).
Each punch can be centered anywhere in the plane and removes all points within distance 1 from its center. What is the minimum number of punches needed to remove every point in the annulus between the circles of radius 7 and 10 (with the same center)? Describe your construction. The person with the smallest number of punches earns +4, and the next four best answers earn +3.
Find all positive integers \( a, b \) such that
\[
\frac{1}{a} + \frac{1}{b} = \frac{p_1}{p_2}
\]
where \( p_1 \) and \( p_2 \) are consecutive primes.
Let \( X_1, X_2, \ldots \) be an infinite sequence of standard normal random variables which are not necessarily independent. Show that there exists a universal constant \( C \) such that \(\mathbb{E} \left[ \max_i \frac{|X_i|}{\sqrt{1 + \log i}} \right] \leq C\).
The best solution was submitted by 채지석 (수리과학과 석박통합과정, +4). Congratulations!
Here is the best solution of problem 2025-11.
Other solutions were submitted by 김동훈 (수리과학과 22학번, +3), 정서윤 (수리과학과 학사과정, +3), Anar Rzayev (수리과학과 19학번, +3).
Let \(P\) be a regular \(2n\)-gon. A perfect matching is a partition of vertex points into \(n\) unordered pairs; each pair represents a chord drawn inside \(P\). Two chords are said to “intersect” if they have a nonempty intersection.
Let \(X\) be the (random) number of intersection points (formed by intersecting chords) in a perfect matching chosen uniformly at random from the set of all possible matchings. Note that more than two chords can intersect at the same point, and in this case this intersection point is just counted once. Compute \(\lim_{n\rightarrow \infty} \frac{\mathbb E[X]}{n^2}\).
The best solution was submitted by Anar Rzayev (수리과학과 19학번, +4). Congratulations!
Here is the best solution of problem 2025-10.
Other solutions were submitted by 김동훈 (수리과학과 22학번, +3), 김준홍 (수리과학과 석박통합과정, +3), 신민규 (수리과학과 24학번, +3), 정서윤 (수리과학과 학사과정, +2).