Prove that for every positive integer \( k \) there exists a positive integer \( n \) such that
\[
\frac{(n+1)(n+2) \dots (2n-k)}{n(n-1) \dots (n-k+1)}
\]
is an integer and that \( k = o(n) \) for such \( n \).
The best solution was submitted by 김은성 (대구과학고등학교, +4). Congratulations!
Here is the best solution of problem 2025-19.
Another solution was submitted by 정영훈 (수리과학과 24학번, +3).
Find all integers \( k \) such that the sequence \( (3n^2 + 3nk^2 + k^3 )_{n=1, 2, \dots} \) contains infinitely many squares.
The best solution was submitted by 김은성 (대구과학고등학교, +4). Congratulations!
Here is the best solution of problem 2025-18.
Other (partial) solutions were submitted by 정영훈 (수리과학과 24학번, +2), Huseyn Ismayilov (전산학부 22학번, +2).
Prove that for every positive integer \( k \) there exists a positive integer \( n \) such that
\[
\frac{(n+1)(n+2) \dots (2n-k)}{n(n-1) \dots (n-k+1)}
\]
is an integer and that \( k = o(n) \) for such \( n \).
Prove the following identity:
\[
\sum_{k=0}^{n-1} \binom{z}{k} \frac{x^{n-k}}{n-k} = \sum_{k=1}^n \binom{z-k}{n-k} \frac{(x+1)^k -1}{k}.
\]
The best solution was submitted by 정서윤 (수리과학과 23학번, +4). Congratulations!
Here is the best solution of problem 2025-17.
Other solutions were submitted by 김은성 (대구과학고, +3), 김찬우 (연세대학교 수학과, +3), 정영훈 (수리과학과 24학번, +3), Huseyn Ismayilov (전산학부 22학번, +3).
Find all integers \( k \) such that the sequence \( (3n^2 + 3nk^2 + k^3 )_{n=1, 2, \dots} \) contains infinitely many squares.
Note: The numbering was wrong. It should be 2025-18.
Show that if \(X\) is a Poisson random variable with parameter \(\mu\), there exists a constant \(c>0\) such that for \(t>\mu+1\), \(\mathbb{P}(X-\mu \geq t)\geq ce^{-2t\log (1+(t+1)/\mu)}\).
The best solution was submitted by Huseyn Ismayilov (전산학부 22학번, +4). Congratulations!
Here is the best solution of problem 2025-16.
Another solution was submitted by 정서윤 (수리과학과 23학번, +3).
Prove the following identity:
\[
\sum_{k=0}^{n-1} \binom{z}{k} \frac{x^{n-k}}{n-k} = \sum_{k=1}^n \binom{z-k}{n-k} \frac{(x+1)^k -1}{k}.
\]
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 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).