Notice on POW 2024-05 and POW 2024-06

It is found that there is a flaw in POW 2024-05; some students showed that the collection of all Knotennullstelle numbers is not a discrete subset of \( \mathbb{C} \). We again apologize for the inconvenience.

To acknowledge the students who reported the flaws in POW 2024-05 and POW 2024-06, we decided to give credits to 김준홍 (KAIST 수리과학과 20학번, +4) and 지은성 (KAIST 수리과학과 20학번, +3) for POW 2024-05 and Anar Rzayev (KAIST 전산학부 19학번, +4) for POW 2024-06.

Here is a “solution” of problem 2024-05.

GD Star Rating
loading...

POW 2024-06 Canceled

It is found that there is a flaw in POW 2024-06; the inequality in the problem is not satisfied with the given g(t). Since it is too late to revise the problem again with a new deadline, we decide to cancel POW 2024-06. We apologize for the inconvenience.

GD Star Rating
loading...

2024-07 Limit of a sequence

For fixed positive numbers \( x_1, x_2, \dots, x_m \), we define a sequence \( \{ a_n \} \) by \( a_n = x_n \) for \(n \leq m \) and
\[
a_n = a_{n-1}^r + a_{n-2}^r + \dots + a_{n-k}^r
\]
for \( n > m \), where \( r \in (0, 1) \). Find \( \lim_{n \to \infty} a_n \).

GD Star Rating
loading...

Notice on POW 2024-06

In POW 2024-06, there was a typo in the assumption. It is now corrected that the assumption holds for \( t \in [-1, 1] \). (Originally, it was for \( t \in \mathbb{R} \).)

GD Star Rating
loading...

2024-06 Limit of concave functions

Let \(f_n(t)\), \(n=1,2…\) be a sequence of concave functions on \(\mathbb{R}\). Assume \(\liminf_{n\to\infty} f_n(t) \geq 2024\,t^{5}+3\) for \(t\in [-1, 1]\) and \(\lim_{n\to \infty} f_n(0) = 3\). Suppose \(f_n'(0)\) exist for \(n=1,2,…\). Compute \(\lim_{n\to \infty} f_n'(0)\).

GD Star Rating
loading...

2024-05 Knotennullstelle

A complex number \(z \in S^1 \smallsetminus \{1\} \) is called a Knotennullstelle if there exists a Laurent polynomial \(p(t) \in \mathbb{Z} [t,t^{-1}]\) such that \(p(1) =\pm 1\) and \(p(z)=0\). Prove that the collection of all Knotennullstelle numbers is a discrete subset of \(\mathbb{C}\).

GD Star Rating
loading...

Solution: 2024-04 Real random variable

Prove the following: There exists a bounded real random variable \( Z \) such that
\[
E[Z] = 0, E[Z^2] = 1, E[Z^3] = x, E[Z^4] = y
\]
if and only if \( y \geq x^2 + 1 \). (Here, \( E \) denotes the expectation.)

The best solution was submitted by 김준홍 (KAIST 수리과학과 20학번, +4). Congratulations!

Here is the best solution of problem 2024-04.

Other solutions were submitted by 신정연 (KAIST 수리과학과 21학번, +3), 채지석 (KAIST 수리과학과 석박통합과정 21학번, +3), 김지원 (KAIST 새내기과정학부 24학번, +2), 김찬우 (연세대학교 수학과 22학번, +2), 박상현 (고려대학교 수학과 20학번, +2), 이명규 (KAIST 전산학부 20학번, +2), 정영훈 (KAIST 새내기과정학부 24학번, +2). There were incorrect solutions submitted.

GD Star Rating
loading...