# 2024-10 Supremum

Find
$\sup \left[ \sum_{n=1}^{\infty} \frac{1}{\sqrt{n}} \left( \sum_{i=n}^{\infty} x_i^2 \right)^{1/2} \Big/ \sum_{i=1}^{\infty} x_i \right],$
where the supremum is taken over all monotone decreasing sequences of positive numbers $$(x_i)$$ such that $$\sum_{i=1}^{\infty} x_i < \infty$$.

GD Star Rating

# Solution: 2024-08 Determinants of 16 by 16 matricies

Let $$A$$ be a $$16 \times 16$$ matrix whose entries are either $$1$$ or $$-1$$. What is the maximum value of the determinant of $$A$$?

The best solution was submitted by 이명규 (KAIST 전산학부 20학번, +4).

Congratulations!

Other solutions were submitted by 김준홍 (KAIST 수리과학과 20학번, +3), 김지원 (KAIST 새내기과정학부 24학번, +3), 신정연 (KAIST 수리과학과 21학번, +3), 정영훈 (KAIST 새내기과정학부 24학번, +3), 지은성 (KAIST 수리과학과 20학번, +3), 채지석 (KAIST 수리과학과 석박통합과정 21학번, +3), Anar Rzayev (KAIST 전산학부 19학번, +3), 권오관 (연세대학교 수학과 22학번, +2).

GD Star Rating

# 2024-09 Integer sums

Find all positive numbers $$a_1,…,a_{5}$$ such that $$a_1^\frac{1}{n} + \cdots + a_{5}^\frac{1}{n}$$ is integer for every integer $$n\geq 1.$$

GD Star Rating

# Solution: 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 $$\lim_{n \to \infty} a_n$$.

The best solution was submitted by 채지석 (KAIST 수리과학과 석박통합과정 21학번, +4). Congratulations!

Other solutions were submitted by 김준홍 (KAIST 수리과학과 20학번, +3), 박지운 (KAIST 새내기과정학부 24학번, +3), 정영훈 (KAIST 새내기과정학부 24학번, +3), Anar Rzayev (KAIST 전산학부 19학번, +2), Sasa Sa (+3).

GD Star Rating

# 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

# 2024-08 Determinants of 16 by 16 matricies

Let $$A$$ be a $$16 \times 16$$ matrix whose entries are either $$1$$ or $$-1$$. What is the maximum value of the determinant of $$A$$?

GD Star Rating

# 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

# 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

# Another notice on POW 2024-06

Due to the change of the assumption in POW 2024-06, the due date for the submitting the solution is postponed to May 13 (Mon.) 3PM. (Originally, it was 3PM Friday.)

GD Star Rating
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 $$t \in \mathbb{R}$$.)