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 \( n \).
Monthly Archives: November 2025
Solution: 2025-17 Combinatorial identity
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).
2025-18 Infinitely many squares
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.
Solution: 2015-16 Poisson variable
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).
2025-17 Combinatorial identity
Prove the following identity:
\[
\sum_{k=0}^{n-1} \binom{z}{k} \frac{x^{n-k}}{n-k} = \sum_{k=1}^n \binom{z-k}{n-k} \frac{(x+1)^k -1}{k}.
\]
Solution: 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 \(P\).
The best solution was submitted by 정영훈 (수리과학과 24학번, +4). Congratulations!
Here is the best solution of problem 2025-15.
2015-16 Poisson variable
Show that if \(X\) is a Poisson random variable with parameter \(\mu\), there exists a constant \(c>0\) such that for \(t>\mu+1\), \(\mathbb{P}(X-\mu \geq t)\geq ce^{-2t\log (1+(t+1)/\mu)}\).