Monthly Archives: November 2023

Solution: 2023-21 A limit

Find the following limit:

\[
\lim_{n \to \infty} \left( \frac{\sum_{k=1}^{n+2} k^k}{\sum_{k=1}^{n+1} k^k} – \frac{\sum_{k=1}^{n+1} k^k}{\sum_{k=1}^{n} k^k} \right)
\]

The best solution was submitted by 문강연 (KAIST 수리과학과 22학번, +4). Congratulations!

Here is the best solution of problem 2023-21.

Other solutions were submitted by 김기수 (KAIST 수리과학과 18학번, +3), 김준홍 (KAIST 수리과학과 20학번, +3), 김찬우 (연세대학교 수학과 22학번, +3), 박기윤 (KAIST 새내기과정학부 23학번, +3), 이도현 (KAIST 수리과학과 석박통합과정 23학번, +3), 이명규 (KAIST 전산학부 20학번, +3), 지은성 (KAIST 수리과학과 20학번, +3), 채지석 (KAIST 수리과학과 석박통합과정 21학번, +3), Adnan Sadik (KAIST 새내기과정학부 23학번, +3), Muhammadfiruz Hasanov (+3), 조현준 (KAIST 수리과학과 22학번, +2), 서성욱 (대전동산고 2학년, +2).

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Solution: 2023-20 A sequence with small tail

Can we find a sequence \(a_i, i=0,1,2,…\) with the following property: for each given integer \(n\geq 0\), we have \[\lim_{L\to +\infty}\sum_{i=0}^L 2^{ni} |a_i|\leq 23^{(n+11)^{10}} \quad \text{ and }\quad \lim_{L\to +\infty}\sum_{i=0}^L 2^{ni} a_i = (-1)^n ?\]

The best solution was submitted by 김기수 (KAIST 수리과학과 18학번, +4). Congratulations!

Here is the best solution of problem 2023-20.

Another solution was submitted by 조현준 (KAIST 수리과학과 22학번, +2).

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2023-21 A limit

Find the following limit:

\[
\lim_{n \to \infty} \left( \frac{\sum_{k=1}^{n+2} k^k}{\sum_{k=1}^{n+1} k^k} – \frac{\sum_{k=1}^{n+1} k^k}{\sum_{k=1}^{n} k^k} \right)
\]

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Solution: 2023-19 Counting the number of solutions

Let \( N \) be the number of ordered tuples of positive integers \( (a_1, a_2, \dots, a_{27}) \) such that \( \frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_{27}} = 1\). Compute the remainder of \( N \) when \( N \) is divided by \(33 \).

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

Here is the best solution of problem 2023-19.

Other solutions were submitted by 강지민 (세마고 3학년, +3), 김기수 (KAIST 수리과학과 18학번, +3), 김민서 (KAIST 수리과학과 19학번, +3), 김준홍 (KAIST 수리과학과 20학번, +3), 김찬우 (연세대학교 수학과 22학번, +3), 박기윤 (KAIST 새내기과정학부 23학번, +3), 이도현 (KAIST 수리과학과 석박통합과정 23학번, +3), 조현준 (KAIST 수리과학과 22학번, +3), 지은성 (KAIST 수리과학과 20학번, +3), 채지석 (KAIST 수리과학과 석박통합과정 21학번, +3), Adnan Sadik (KAIST 새내기과정학부 23학번, +3), Dzhamalov Omurbek (KAIST 전산학부 22학번, +3), Kharchenka Yuliya (KAIST 물리학과 22학번, +3), Muhammadfiruz Hasanov (+3), Aiden Stock (+3).

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2023-20 A sequence with small tail

Can we find a sequence \(a_i, i=0,1,2,…\) with the following property: for each given integer \(n\geq 0\), we have \[\lim_{L\to +\infty}\sum_{i=0}^L 2^{ni} |a_i|\leq 23^{(n+11)^{10}} \quad \text{ and }\quad \lim_{L\to +\infty}\sum_{i=0}^L 2^{ni} a_i = (-1)^n ?\]

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Solution: 2023-18 Degrees of a graph

Find all integers \( n \geq 8 \) such that there exists a simple graph with \( n \) vertices whose degrees are as follows:

(i) \( (n-4) \) vertices of the graph are with degrees \( 4, 5, 6, \dots, n-2, n-1 \), respectively.

(ii) The other \( 4 \) vertices are with degrees \( n-2, n-2, n-1, n-1 \), respectively.

The best solution was submitted by 이도현 (KAIST 수리과학과 석박통합과정 23학번, +4). Congratulations!

Here is the best solution of problem 2023-18.

Other solutions were submitted by 강지민 (세마고 3학년, +3), 김기수 (KAIST 수리과학과 18학번, +3), 김민서 (KAIST 수리과학과 19학번, +3), 김찬우 (연세대학교 수학과 22학번, +3), 나경민 (KAIST 전산학부 22학번, +3), 박기윤 (KAIST 새내기과정학부 23학번, +3), 이명규 (KAIST 전산학부 20학번, +3), 전해구 (KAIST 기계공학과 졸업생, +3), 조현준 (KAIST 수리과학과 22학번, +3), 지은성 (KAIST 수리과학과 20학번, +3), 채지석 (KAIST 수리과학과 석박통합과정 21학번, +3), 최민규 (한양대학교 의과대학 졸업생, +3), Adnan Sadik (KAIST 새내기과정학부 23학번, +3), Dzhamalov Omurbek (KAIST 전산학부 22학번, +3), Muhammadfiruz Hasanov (+3).

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