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.).

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).

Let \(f:\mathbb{R}^n\to \mathbb{R}^{n-1}\) be a function such that for each point a in \(\mathbb{R}^n\), the limit $$\lim_{x\to a} \frac{|f(x)-f(a)|}{|x-a|}$$ exists. Prove that f is a constant function.

The best solution was submitted by Minjae Park (박민재), 2011학번. Congratulations!

Here is his Solution of Problem 2011-23.

Let p be a prime number and let n be a positive integer. Let \(A=\left( \binom{i+j-2}{i-1}\right)_{1\le i\le p^n, 1\le j\le p^n} \) be a \(p^n \times p^n\) matrix. Prove that \( A^3 \equiv I \pmod p\), where I is the \(p^n \times p^n\) identity matrix.

The best solution was submitted by Minjae Park (박민재), 2011학번. Congratulations!

Here is his Solution of Problem 2012-6.

Alternative solutions were submitted by 서기원 (수리과학과 2009학번, +3), 이명재 (2012학번, +2).

For given positive real numbers \(a_1,\ldots,a_k\) and for each integer n≥k, let \(a_{n+1}\) be the geometric mean of \( a_n, a_{n-1}, a_{n-2}, \ldots, a_{n-k+1}\). Prove that \( \lim_{n\to\infty} a_n\) exists and compute this limit.

The best solution was submitted by Gee Won Suh (서기원), 수리과학과 2009학번. Congratulations!

Here is his Solution of Problem 2012-5.

Alternative solutions were submitted by 박민재 (2011학번, +3, Solution), 김태호 (2011학번, +3, Solution), 이명재 (2012학번, +3), 박훈민 (대전과학고등학교 2학년, +3), 윤영수 (2011학번, +2), 조준영 (2012학번, +2), 변성철 (2011학번, +2), 정우석 (서강대학교 자연과학부 2011학번, +2). One incorrect solution was received.

Find the smallest and the second smallest odd integers n satisfying the following property: \[ n=x_1^2+y_1^2 \text{ and } n^2=x_2^2+y_2^2 \] for some positive integers \(x_1,y_1,x_2,y_2\) such that \(x_1-y_1=x_2-y_2\).

The best solution was submitted by Minjae Park (박민재), 2011학번. Congratulations!

Here is his Solution of Problem 2012-4.

Alternative solutions were submitted by 조준영 (2012학번, +3), 서기원 (수리과학과 2009학번, +3), 임창준 (2012학번, +3), 홍승한 (2012학번, +2), 이명재 (2012학번, +2), 김현수 (?, +3), 천용 (전남대, +2). One incorrect solution was received.

Compute \[ f(x)= \int_{0}^1 \frac{\log (1- 2t\cos x + t^2) }{t} dt.\]

The best solution was submitted by Younghun Lee (이영훈), 2011학번.

Here is his Solution of Problem 2012-3.

Alternative solutions were submitted by 조준영 (2012학번, +3, Solution), 김태호 (2011학번, +3), 서기원 (수리과학과 2009학번, +3), 박민재 (2011학번, +3, Solution), 이명재 (2012학번, +3), 서동휘 (수리과학과 2009학번, +2), 임정환 (수리과학과 2009학번, +2), 김현수 (?, +2), 정우석 (서강대 자연과학부 2011학번, +2), 조위지 (Stanford Univ. 물리학과 박사과정, +3, Solution).