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.
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 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.
Good luck next week for your midterm exam! We take break and return on March 30, Friday.
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).
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.
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.
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.
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 \(x_1-y_1=x_2-y_2\).