Concluding POW 2024 spring semester

POW 2024 spring semester has ended. We apologize for many issues we had experienced this semester. Thank you for your participation, and see you in the fall semester.

GD Star Rating

Solution: 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 $$\sum_{i=1}^{\infty} x_i < \infty$$.

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

There were incorrect solutions submitted.

GD Star Rating

Solution: 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 $$n\geq 1.$$

The best solution was submitted by 권오관 (연세대학교 수학과 22학번, +4). Congratulations!

Other solutions were submitted by 김준홍 (KAIST 수리과학과 20학번, +3), 김지원 (KAIST 새내기과정학부 24학번, +3), 박지운 (KAIST 새내기과정학부 24학번, +3), 신정연 (KAIST 수리과학과 21학번, +3), 이명규 (KAIST 전산학부 20학번, +3), 정영훈 (KAIST 새내기과정학부 24학번, +3), 채지석 (KAIST 수리과학과 석박통합과정 21학번, +3), Anar Rzayev (KAIST 전산학부 19학번, +3).

GD Star Rating

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