Compute \(\int_0^1 \frac{x^k-1}{\log x}dx\).

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

Here is his Solution of Problem 2012-22.

Alternative solutions were submitted by 박민재 (2011학번, +3), 서기원 (수리과학과 2009학번, +2), 김태호 (수리과학과 2011학번, +2), 임현진 (물리학과 2010학번, +2), 조위지 (Stanford Univ. 물리학과 박사과정, +3), 박훈민 (대전과학고 2학년, +3).

Let \(n\) be a fixed positive integer and let \(p\in (0,1)\). Let \(D_n\) be the determinant of a random \(n\times n\) 0-1 matrix whose entries are independent identical random variables, each of which is 1 with the probability \(p\) and 0 with the probability \(1-p\).  Find the expected value and variance of \(D_n\).

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

Here is his Solution of Problem 2012-21.

Alternative solutions were submitted by 박민재 (2011학번, +3), 김태호 (수리과학과 2011학번, +3), 임현진 (물리학과 2010학번, +3), 김지홍 (수리과학과 2007학번, +2), 서기원 (수리과학과 2009학번, +2).

Let \(a_0=3\) and \(a_{n}=a_{n-1}+\sqrt{a_{n-1}^2+3}\) for all \(n\ge 1\). Determine \[\lim_{n\to\infty}\frac{a_n}{2^n}.\]

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

Here is his Solution of Problem 2012-19.

Alternative solutions were submitted by 박민재 (2011학번, +3), 김태호 (수리과학과 2011학번, +3). Two incorrect solutions were submitted (YSC, KJW).

Prove that if a finite ring has two elements \(x\) and \(y\) such that \(xy^2=y\), then \( yxy=y\).

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

Here is Solution of Problem 2012-16.

Alternative solutions were submitted by 김주완 (수리과학과 2010학번, +3), 김지원 (수리과학과 2010학번, +3), 서기원 (수리과학과 2009학번, +3), 김태호 (수리과학과 2011학번, +3), 임현진 (물리학과 2010학번, +3), 박민재 (2011학번, +3), 조상흠 (수리과학과 2010학번, +3), 정우석 (서강대 수학과 2011학번, +3). One incorrect solution (KHK) was submitted.

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.

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

Here is his Solution of Problem 2012-11.

Alternative solution was submitted by 박민재(2011학번, +3). Two incorrect solutions were submitted (W.S.J., K.M.P.).

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.).

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).

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.).