Monthly Archives: October 2025

2025-15 locally Lipschitz functions

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 \(P\).

Solution: 2025-14 Convex hulls

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).

Solution: 2025-13 Covering the Donut with Unit Punches

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).