# Solution: 2012-10 Platonic solids

Determine all Platonic solids that can be drawn with the property that all of its vertices are rational points.

The best solution was submitted by Myeongjae Lee (이명재), 2012학번. Congratulations!

Here is his Solution of Problem 2012-10.

Alternative solutions were submitted by 박민재 (2011학번, +3, Solution), 정우석 (서강대학교 2011학번, +3), 박훈민 (대전과학고 2학년, +2). One incorrect solution was submitted (G.S.).

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# 2012-11 Dividing a circle

Let f be a continuous function from [0,1] such that f([0,1]) is a circle. Prove that there exists two closed intervals $$I_1, I_2 \subseteq [0,1]$$ such that $$I_1\cap I_2$$ has at most one point, $$f(I_1)$$ and $$f(I_2)$$ are semicircles, and $$f(I_1)\cup f(I_2)$$ is a circle.

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# Solution: 2012-9 Rank of a matrix

Let M be an n⨉n matrix over the reals. Prove that $$\operatorname{rank} M=\operatorname{rank} M^2$$ if and only if $$\lim_{\lambda\to 0} (M+\lambda I)^{-1}M$$ exists.

The best solution was submitted by Myeongjae Lee (이명재), 2012학번. Congratulations!

Here is his Solution of Problem 2012-9.

Alternative solutions were submitted by 서기원 (수리과학과 2009학번, +3), 박민재 (2011학번, +3).

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# 2012-9 Rank of a matrix

Let M be an n⨉n matrix over the reals. Prove that $$\operatorname{rank} M=\operatorname{rank} M^2$$ if and only if $$\lim_{\lambda\to 0} (M+\lambda I)^{-1}M$$ exists.

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# Solution: 2012-8 Non-fixed points

Let X be a finite non-empty set. Suppose that there is a function $$f:X\to X$$ such that $$f^{20120407}(x)=x$$ for all $$x\in X$$. Prove that the number of elements x in X such that $$f(x)\neq x$$ is divisible by 20120407.

The best solution was submitted by Myeongjae Lee (이명재), 2012학번. Congratulations!

Here is his Solution of Problem 2012-8.

Alternative solutions were submitted by Phan Kieu My (전산학과 2009학번, +3), 김태호 (수리과학과 2011학번, +3), 박민재 (2011학번, +3), 서기원 (수리과학과 2009학번, +3), 천용 (전남대 의예과 2011학번, +3), 어수강 (서울대학교 석사과정, +3), 정우석 (서강대학교 2011학번, +3), 박훈민 (대전과학고 2학년, +3). There were 2 incorrect solutions (S. B., S. H.).

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# Solution of 2012-7: Product of Sine

Let X be the set of all postive real numbers c such that  $\frac{\prod_{k=1}^{n-1} \sin\left( \frac{k \pi}{2n}\right)}{c^n}$  converges as n goes to infinity. Find the infimum of X.

The best solution was submitted by Taeho Kim (김태호, 수리과학과 2011학번). Congratulations!

Here is his Solution of Problem 2012-7.

Alternative solutions were submitted by 서기원 (수리과학과 2009학번, +3), 박민재 (2011학번, +3), 조준영 (2012학번, +3), 이명재 (2012학번, +3), 정우석 (서강대 2011학번, +3), 천용 (전남대 의예과 2011학번 +3), 어수강 (서울대학교 석사과정, +2).

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Let X be a finite non-empty set. Suppose that there is a function $$f:X\to X$$ such that $$f^{20120407}(x)=x$$ for all $$x\in X$$. Prove that the number of elements x in X such that $$f(x)\neq x$$ is divisible by 20120407.