# 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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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.