Let f(n) be the largest integer k such that n! is divisible by \(n^k\). Prove that \[ \lim_{n\to \infty} \frac{(\log n)\cdot \max_{2\le i\le n} f(i)}{n \log\log n}=1.\]
The best solution was submitted by Minjae Park (박민재), 2011학번. Congratulations!
Here is his Solution of Problem 2011-7.
Alternative solutions were submitted by 양해훈 (수리과학과 2008학번, +3), 이재석 (수리과학과 2007학번, +2).
Problem 2011-11 was the last problem of this semester. Good luck to your final exam! We wish you to come back in the fall semester. We will start in the first week of September.
Soon, we will have a small ceremony to award winners.
Prove that for every skew-symmetric matrix A, there are symmetric matrices B and C such that A=BC-CB.
The best solution was submitted by Minjae Park (박민재), 2011학번. Congratulations!
Here is his Solution of Problem 2011-11.
Alternative solutions were submitted by 강동엽 (전산학과 2009학번, +3), 서기원 (수리과학과 2009학번, +3), 어수강 (홍익대 수학교육과, +3, Alternative Solution of Problem 2011-11).
Let \(t_1,t_2,\ldots,t_n\) be positive integers. Let \(p(x_1,x_2,\dots,x_n)\) be a polynomial with n variables such that \(\deg(p)\le t_1+t_2+\cdots+t_n\). Prove that \(\left(\frac{\partial}{\partial x_1}\right)^{t_1} \left(\frac{\partial}{\partial x_2}\right)^{t_2}\cdots \left(\frac{\partial}{\partial x_n}\right)^{t_n} p\) is equal to \[\sum_{a_1=0}^{t_1} \sum_{a_2=0}^{t_2}\cdots \sum_{a_n=0}^{t_n} (-1)^{t_1+t_2+\cdots+t_n+a_1+a_2+\cdots+a_n}\left( \prod_{i=1}^n \binom{t_i}{a_i} \right)p(a_1,a_2,\ldots,a_n).\]
The best solution was submitted by Kang, Dongyub (강동엽), 전산학과 2009학번. Congratulations!
Here is his Solution of Problem 2011-10
An alternative solution was submitted by 박민재 (2011학번, +3).
Prove that there is a constant c>1 such that if \(n>c^k\) for positive integers n and k, then the number of distinct prime factors of \(n \choose k\) is at least k.
The best solution was submitted by Minjae Park (박민재), KAIST 2011학번. Congratulations!
Here is his Solution of Problem 2011-9.
An alternative solution was submitted by 어수강 (홍익대 수학교육과 2004학번, +3).
Prove that for every skew-symmetric matrix A, there are symmetric matrices B and C such that A=BC-CB.
Let \(t_1,t_2,\ldots,t_n\) be positive integers. Let \(p(x_1,x_2,\dots,x_n)\) be a polynomial with n variables such that \(\deg(p)\le t_1+t_2+\cdots+t_n\). Prove that \(\left(\frac{\partial}{\partial x_1}\right)^{t_1} \left(\frac{\partial}{\partial x_2}\right)^{t_2}\cdots \left(\frac{\partial}{\partial x_n}\right)^{t_n} p\) is equal to \[\sum_{a_1=0}^{t_1} \sum_{a_2=0}^{t_2}\cdots \sum_{a_n=0}^{t_n} (-1)^{t_1+t_2+\cdots+t_n+a_1+a_2+\cdots+a_n}\left( \prod_{i=1}^n \binom{t_i}{a_i} \right)p(a_1,a_2,\ldots,a_n).\]
Prove that there is a constant c>1 such that if \(n>c^k\) for positive integers n and k, then the number of distinct prime factors of \(n \choose k\) is at least k.
Let f be a continuous function on [0,1]. Prove that \[ \lim_{n\to \infty}\int_0^1 \cdots \int_0^1 f(\sqrt[n]{x_1 x_2 \cdots x_n } ) dx_1 dx_2 \cdots dx_n = f(1/e).\]
The best solution was submitted by Kang, Dongyub (강동엽), 전산학과 2009학번. Congratulations!
Here is his Solution of Problem 2011-8.
Alternative solutions were submitted by 김치헌 (수리과학과 2006학번, +3), 어수강 (홍익대학교 수학교육과 2004학번, +3).
(Here is a Solution by Chiheon Kim for Problem 2011-8.)
Let f be a continuous function on [0,1]. Prove that \[ \lim_{n\to \infty}\int_0^1 \cdots \int_0^1 f(\sqrt[n]{x_1 x_2 \cdots x_n } ) dx_1 dx_2 \cdots dx_n = f(1/e).\]