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.