# 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 $$A$$.)

(This is the last problem of this semester. Thank you for participating KAIST Math Problem of the Week.)

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# Solution: 2015-11 Limit

Does $$\frac{1}{n \sin n}$$ converge as $$n$$ goes to infinity?

The best solution was submitted by Lee, Jongwon (이종원, 수리과학과 2014학번). Congratulations!

Here is his solution of problem 2015-11.

Alternative solutions were submitted by 고경훈 (2015학번, +3), 김기현 (수리과학과 2012학번, +3), 신준형 (2015학번, +3), 엄태현 (수리과학과 2012학번, +3), 오동우 (2015학번, +3), 이수철 (수리과학과 2012학번, +3), 진우영 (수리과학과 2012학번, +3), 함도규 (2015학번, +3), 이상민 (수리과학과 2014학번, +2), 이영민 (수리과학과 2012학번, +2). One incorrect solution (KDR) was submitted.

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# Solution: 2015-10 Product of sine functions

Let $$w_1,w_2,\ldots,w_n$$ be positive real numbers such that $$\sum_{i=1}^n w_i=1$$. Prove that if $$x_1,x_2,\ldots,x_n\in [0,\pi]$$, then $\sin \left(\prod_{i=1}^n x_i^{w_i} \right) \ge \prod_{i=1}^n (\sin x_i)^{w_i}.$

The best solution was submitted by Lee, Young Min (이영민, 수리과학과 2012학번). Congratulations!

Here is his solution of problem 2015-10.

Other (but mostly identical) solutions were submitted by 고경훈 (2015학번, +3), 김기현 (수리과학과 2012학번, +3), 엄태현 (수리과학과 2012학번, +3), 오동우 (2015학번, +3), 이수철 (수리과학과 2012학번, +3), 이종원 (수리과학과 2014학번, +3), 진우영 (수리과학과 2012학번, +3), 함도규 (2015학번, +3).

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# 2015-11 Limit

Does $$\frac{1}{n \sin n}$$ converge as $$n$$ goes to infinity?

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# Solution: 2015-9 Sum of squares

Let $$n\ge 1$$ and $$a_0,a_1,a_2,\ldots,a_{n}$$ be non-negative integers. Prove that if $N=\frac{a_0^2+a_1^2+a_2^2+\cdots+a_{n}^2}{1+a_0a_1a_2\cdots a_{n}}$ is an integer, then $$N$$ is the sum of $$n$$ squares of integers.

The best solution was submitted by Lee, Jongwon (이종원, 수리과학과 2014학번). Congratulations!

Here is his solution of problem 2015-9.

Alternative solutions were submitted by 김기현 (수리과학과 2012학번, +3), 엄태현 (수리과학과 2012학번, +3), 이수철 (수리과학과 2012학번, +3), 진우영 (수리과학과 2012학번, +3), 함도규 (2015학번, +3), 윤지훈 (2012학번, +2). One incorrect solution was submitted (YSC).

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# 2015-10 Product of sine functions

Let $$w_1,w_2,\ldots,w_n$$ be positive real numbers such that $$\sum_{i=1}^n w_i=1$$. Prove that if $$x_1,x_2,\ldots,x_n\in [0,\pi]$$, then $\sin \left(\prod_{i=1}^n x_i^{w_i} \right) \ge \prod_{i=1}^n (\sin x_i)^{w_i}.$

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# Solution: 2015-8 all lines

Does there exist a subset $$A$$ of $$\mathbb{R}^2$$ such that $$\lvert A\cap L\rvert=2$$ for every straight line $$L$$?

The best solution was submitted by Lee, Su Cheol (이수철, 수리과학과 2012학번). Congratulations!

Here is his solution of problem 2015-08.

Alternative solutions were submitted by 김기현 (수리과학과 2012학번, +3), 김동률 (2015학번, +3), 엄태현 (수리과학과 2012학번, +3), 이종원 (수리과학과 2014학번, +3), 진우영 (수리과학과 2012학번, +3), 박훈민 (수리과학과 2013학번, +2), 오동우 (2015학번, +2).

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# 2015-9 Sum of squares

Let $$n\ge 1$$ and $$a_0,a_1,a_2,\ldots,a_{n}$$ be non-negative integers. Prove that if $N=\frac{a_0^2+a_1^2+a_2^2+\cdots+a_{n}^2}{1+a_0a_1a_2\cdots a_{n}}$ is an integer, then $$N$$ is the sum of $$n$$ squares of integers.

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# 2015-8 all lines

Does there exist a subset $$A$$ of $$\mathbb{R}^2$$ such that $$\lvert A\cap L\rvert=2$$ for every straight line $$L$$?

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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 $$0\le r\le n-k-1$$.