# 2015-7 Binomial Identity

Prove or disprove that $\sum_{i=0}^r (-1)^i \binom{i+k}{k} \binom{n}{r-i} = \binom{n-k-1}{r}$ if $$k, r$$ are non-negative integers and $$0\le r\le n-k-1$$.

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# Solution: 2015-6 Dense sets

Let $$A$$ be an unbounded subset of the set $$\mathbb R$$ of the real numbers. Let $$T$$ be the set of all real numbers $$t$$ such that $$\{tx-\lfloor tx\rfloor : x\in A\}$$ is dense in $$[0,1]$$. Is $$T$$ dense in $$\mathbb R$$?

The best solution was submitted by Kim, Kihyun (김기현, 수리과학과 2012학번). Congratulations!

Here is his solution of problem 2015-6.

Alternative solutions were submitted by 엄태현 (수리과학과 2012학번, +3), 이명재 (수리과학과 2012학번, +3), 이수철 (수리과학과 2012학번, +3), 이종원 (수리과학과 2014학번, +3), 정성진 (수리과학과 2013학번, +3), 최인혁 (2015학번, +3), 진우영 (수리과학과 2012학번, +3), 배형진 (마포고 1학년, +2). One incorrect solution was submitted (KDR).

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# Midterm break

The problem of the week will take a break during the midterm exam period and return on April 24, Friday. Good luck on your midterm exams!

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# Solution: 2015-5 trace and matrices

Determine all $$n\times n$$ matrices A such that $$\operatorname{tr}(AXY)=\operatorname{tr}(AYX)$$ for all $$n\times n$$ matrices $$X$$ and $$Y$$.

The best solution was submitted by Choi, Doo Seong (최두성, 수리과학과 2011학번). Congratulations!

Here is his solution of problem 2015-5.

Alternative solutions were submitted by 고경훈 (2015학번, +3), 김기현 (수리과학과 2012학번, +3), 김경석 (2015학번, +3), 엄태현 (수리과학과 2012학번, +3), 오동우 (2015학번, +3), 유찬진 (2015학번, +3), 이수철 (수리과학과 2012학번, +3), 이명재 (수리과학과 2012학번, +3), 이영민 (수리과학과 2012학번, +3), 장기정 (수리과학과 2014학번, +3), 정성진 (수리과학과 2013학번, +3), 진우영 (수리과학과 2012학번, +3), 최인혁 (2015학번, +3), 함도규 (2015학번, +3), 홍혁표 (수리과학과 2013학번, +3), 박성혁 (수리과학과 2014학번, +2), 이종원 (수리과학과 2014학번, +2), 전한솔 (고려대, +3), 어수강 (서울대 수리과학부 대학원생, +3).

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# 2015-6 Dense sets

Let $$A$$ be an unbounded subset of the set $$\mathbb R$$ of the real numbers. Let $$T$$ be the set of all real numbers $$t$$ such that $$\{tx-\lfloor tx\rfloor : x\in A\}$$ is dense in $$[0,1]$$. Is $$T$$ dense in $$\mathbb R$$?

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# Solution: 2015-4 An inequality on positive semidefinite matrices

Let $$M=\begin{pmatrix} A & B \\ B^*& C \end{pmatrix}$$ be a positive semidefinite Hermian matrix. Prove that $\operatorname{rank} M \le \operatorname{rank} A +\operatorname{rank} C.$ (Here, $$A$$, $$B$$, $$C$$ are matrices.)

The best solution was submitted by 엄태현 (수리과학과 2012학번). Congratulations!

Here is his solution of problem 2015-04.

Alternative solutions were submitted by 고경훈 (2015학번, +3), 김경석 (2015학번, +3), 김기현 (수리과학과 2012학번, +3), 박성혁 (수리과학과 2014학번, +3), 오동우 (2015학번, +3), 이명재 (수리과학과 2012학번, +3), 이수철 (수리과학과 2012학번, +3, solution), 이종원 (수리과학과 2014학번, +3, solution), 장기정 (수리과학과 2014학번, +3), 정성진 (수리과학과 2013학번, +3), 진우영 (수리과학과 2012학번, +3), 최인혁 (2015학번, +3).

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Determine all $$n\times n$$ matrices A such that $$\operatorname{tr}(AXY)=\operatorname{tr}(AYX)$$ for all $$n\times n$$ matrices $$X$$ and $$Y$$.