# 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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