Solution: 2023-23 Don’t be negative!

Consider a function \(f: \{1,2,\dots, n\}\rightarrow \mathbb{R}\) satisfying the following for all \(1\leq a,b,c \leq n-2\) with \(a+b+c\leq n\).

\[ f(a+b)+f(a+c)+f(b+c) – f(a)-f(b)-f(c)-f(a+b+c) \geq 0 \text{ and } f(1)=f(n)=0.\]

Prove or disprove this: all such functions \(f\) always have only nonnegative values on its domain.

Acknowledgement: This problem arises during a research discussion between June Huh, Jaehoon Kim and Matt Larson.

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

Here is the best solution of problem 2023-23.

Other solutions were submitted by 김기수 (KAIST 수리과학과 18학번, +3), 김찬우 (연세대학교 수학과 22학번, +3), 박기윤 (KAIST 새내기과정학부 23학번, +3), 지은성 (KAIST 수리과학과 20학번, +3), 이도현 (KAIST 수리과학과 석박통합과정 23학번, +3), 전해구 (KAIST 기계공학과 졸업생, +3).

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Solution: 2023-22 Simultaneously diagonalizable matrices

Does there exist a nontrivial subgroup \(G\) of \( GL(10, \mathbb{C}) \) such that each element in \(G\) is diagonalizable but the set of all the elements of \(G\) is not simultaneously diagonalizable?

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

Here is the best solution of problem 2023-22.

Other solutions were submitted by 김기수 (KAIST 수리과학과 18학번, +3), 박기윤 (KAIST 새내기과정학부 23학번, +3), 지은성 (KAIST 수리과학과 20학번, +3), 채지석 (KAIST 수리과학과 석박통합과정 21학번, +3), 이명규 (KAIST 전산학부 20학번, +2).

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2023-23 Don’t be negative!

Consider a function \(f: \{1,2,\dots, n\}\rightarrow \mathbb{R}\) satisfying the following for all \(1\leq a,b,c \leq n-2\) with \(a+b+c\leq n\).

\[ f(a+b)+f(a+c)+f(b+c) – f(a)-f(b)-f(c)-f(a+b+c) \geq 0 \text{ and } f(1)=f(n)=0.\]

Prove or disprove this: all such functions \(f\) always have only nonnegative values on its domain.

Acknowledgement: This problem arises during a research discussion between June Huh, Jaehoon Kim and Matt Larson.

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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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