# 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!

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!

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!

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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# 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?

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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!

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!

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!

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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