# Solution: 2021-13 Not convex

Prove or disprove the following:

There exist an infinite sequence of functions $$f_n: [0, 1] \to \mathbb{R} , n=1, 2, \dots$$ ) such that

(1) $$f_n(0) = f_n(1) = 0$$ for any $$n$$,

(2) $$f_n(\frac{a+b}{2}) \leq f_n(a) + f_n(b)$$ for any $$a, b \in [0, 1]$$,

(3) $$f_n – c f_m$$ is not identically zero for any $$c \in \mathbb{R}$$ and $$n \neq m$$.

The best solution was submitted by 김기택 (수리과학과 대학원생, +4). Congratulations!

Other solutions were submitted by 강한필 (전산학부 2016학번, +3), 고성훈 (수리과학과 2018학번, +3), 김민서 (수리과학과 2019학번, +3), 박정우 (수리과학과 2019학번, +3), 신주홍 (수리과학과 2020학번, +3), 이도현 (수리과학과 2018학번, +3), 이본우 (수리과학과 2017학번, +3), 이호빈 (수리과학과 대학원생, +3).

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# Solution: 2021-10 Integral inequality

Let $$f: [0, 1] \to \mathbb{R}$$ be a continuous function satisfying
$\int_x^1 f(t) dt \geq \int_x^1 t\, dt$
for all $$x \in [0, 1]$$. Prove that
$\int_0^1 [f(t)]^2 dt \geq \int_0^1 t f(t) dt.$

The best solution was submitted by 김기택 (2021학번, +4). Congratulations!

Other solutions were submitted by 강한필 (전산학부 2016학번, +3), 고성훈 (수리과학과 2018학번, +3), 최백규 (생명과학과 대학원, +3).

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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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# Solution: 2015-12 Rank

Let $$A$$ be an $$n\times n$$ matrix with complex entries. Prove that if $$A^2=A^*$$, then $\operatorname{rank}(A+A^*)=\operatorname{rank}(A).$ (Here, $$A^*$$ is the conjugate transpose of $$A$$.)

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

Here is his solution of problem 2015-12.

Alternative solutions were submitted by 고경훈 (2015학번, +3), 김기현 (수리과학과 2012학번, +3), 박지민 (전산학과 대학원생, +3), 엄태현 (수리과학과 2012학번, +3), 이수철 (수리과학과 2012학번, +3), 이종원 (수리과학과 2014학번, +3), 진우영 (수리과학과 2012학번, +3), 함도규 (2015학번, +3).

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