Denote \(P = \{(x, y, z) \in \mathbb{R^3}: 10< x,y,z <31\}\). Suppose a function \(f (v): \mathbb{R^3} \to \mathbb{R_{\geq 0}}\) satisfies:
(a) \(f(\lambda v) = \lambda^{25} f(v)\) for all \(v\in P\) and \(0<\lambda \in \mathbb{R}\),
(b) \(f(v+w) \geq f(v)\) for every \(v, w \in P\),
(c) \(f (v)\) is locally bounded.
Show that \(f (v)\) is locally Lipschitz in \(P\).
The best solution was submitted by 정영훈 (수리과학과 24학번, +4). Congratulations!
Here is the best solution of problem 2025-15.
Show that if \(X\) is a Poisson random variable with parameter \(\mu\), there exists a constant \(c>0\) such that for \(t>\mu+1\), \(\mathbb{P}(X-\mu \geq t)\geq ce^{-2t\log (1+(t+1)/\mu)}\).
Denote \(P = \{(x, y, z) \in \mathbb{R^3}: 10< x,y,z <31\}\). Suppose a function \(f (v): \mathbb{R^3} \to \mathbb{R_{\geq 0}}\) satisfies:
(a) \(f(\lambda v) = \lambda^{25} f(v)\) for all \(v\in P\) and \(0<\lambda \in \mathbb{R}\),
(b) \(f(v+w) \geq f(v)\) for every \(v, w \in P\),
(c) \(f (v)\) is locally bounded.
Show that \(f (v)\) is locally Lipschitz in \(P\).
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.