# 2016-7 Sum-free

For a set $$A \subset \mathbb{R}$$, let $$f(A)$$ be the size of the largest set $$B \subset A$$ such that $$(B+B) \cap B = \emptyset$$. For a positive integer $$n$$, let $$f(n) = \min_{0 \notin A, |A|=n} f(A)$$. Prove that $$f(n) \geq n/3$$.

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# Solution: 2016-6 Convex function

Suppose that $$f$$ is a real-valued convex function on $$\mathbb{R}$$. Prove that the function $$X \mapsto \mathrm{Tr } f(X)$$ on the vector space of $$N \times N$$ Hermitian matrices is convex.

The best solution was submitted by Kook, Yun Bum (국윤범, 수리과학과 2015학번). Congratulations!

Here is his solution of problem 2016-6.

Alternative solutions were submitted by 이종원 (수리과학과 2014학번, +3), 이준호 (2016학번, +3), 장기정 (수리과학과 2014학번, +3), 유현우 (한양대학교 화학공학과 2013학번, +3), 이시우 (포항공대 수학과 2013학번, +3).

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# Midterm break

The problem of the week will take a break during the midterm exam period and return on April 29, Friday. Good luck on your midterm exams!

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# 2016-5 Partition into 4 sets: remains open

There were 3 submissions for problem 2016-5 but no correct solutions were submitted so far.

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# 2016-6 Convex function

Suppose that $$f$$ is a real-valued convex function on $$\mathbb{R}$$. Prove that the function $$X \mapsto \mathrm{Tr } f(X)$$ on the vector space of $$N \times N$$ Hermitian matrices is convex.

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# Solution: 2016-4 Distances in a tree

Let $$T$$ be a tree on $$n$$ vertices $$V=\{1,2,\ldots,n\}$$. For two vertices $$i$$ and $$j$$, let $$d_{ij}$$ be the distance between $$i$$ and $$j$$, that is the number of edges in the unique path from $$i$$ to $$j$$. Let $$D_T(x)=(x^{d_{ij}})_{i,j\in V}$$ be the $$n\times n$$ matrix. Prove that $\det (D_T(x))=(1-x^2)^{n-1}.$

The best solution was submitted by Kim, Kee Tack (김기택, 수리과학과 2015학번). Congratulations!

Here is his solution of problem 2016-4.

Alternative solutions were submitted by 강한필 (2016학번, +3), 국윤범 (수리과학과 2015학번, +3), 김강식 (포항공대 수학과 2013학번, +3), 김경석 (연세대학교 의예과 2016학번, +3), 김동률 (수리과학과 2015학번, +3), 김재현 (2016학번, +3), 박기연 (2016학번, +3), 송교범 (서대전고등학교 3학년, +3), 이시우 (포항공대 수학과 2013학번, +3), 이종원 (수리과학과 2014학번, +3), 이준호 (2016학번, +3), 장기정 (수리과학과 2014학번, +3), 조태혁 (수리과학과 2014학번, +3), Muhammaadfiruz Hasanov (2014학번, +3), 김동규 (수리과학과 2015학번, +2), 김홍규 (수리과학과 2011학번, +2), 배형진 (마포고등학교 2학년, +2), 어수강 (서울대학교 수학교육과 박사과정, +2), 유찬진 (수리과학과 2015학번, +2), 윤준기 (전기및전자공학부 2014학번, +2), 이상민 (수리과학과 2014학번, +2), 이정환 (수리과학과 2015학번, +2).

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Let $$A_1,A_2,\ldots,A_n$$ be subsets of $$\{1,2,\ldots,n\}$$ such that $$i\notin A_i$$ for all $$i$$. Prove that there exist four sets $$C_1,C_2,C_3,C_4$$ such that $$C_1\cup C_2\cup C_3\cup C_4=\{1,2,\ldots,n\}$$ and for all $$i$$ and $$j$$, if $$i\in C_j$$, then $$\lvert A_i\cap C_j\rvert \le \frac12 \lvert A_i\rvert$$.