Concluding 2011 Spring

Thanks all for participating POW actively. Here’s the list of winners:

1st prize: Park, Minjae (박민재) – 2011학번

2nd prize: Kang, Dongyub (강동엽) – 전산학과 2009학번

3rd prize: Suh, Gee Won (서기원) – 수리과학과 2009학번
3rd prize: Lee, Jaeseok (이재석) – 수리과학과 2007학번

Congratulations!

In addition to these three people, I selected one more student to receive one notebook.

Kim, Ji Won (김지원) -수리과학과 2010학번

박민재 (2011학번) 31pts
강동엽 (2009학번) 24pts
서기원 (2009학번) 16pts
이재석 (2007학번) 16pts
김지원 (2010학번) 12pts
김치헌 (2006학번) 5pts
김인환 (2010학번) 3pts
김태호 (2011학번) 3pts
양해훈 (2008학번) 3pts
이동민 (2009학번) 2pts

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Solution: 2011-7 Factorial

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).

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Thanks for participating POW; We will resume on Sep. 2011

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.

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Solution: 2011-11 Skew-symmetric and symmetric matrices

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).

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Solution: 2011-10 Multivariable polynomial

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).

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Solution: 2011-9 Distinct prime factors

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).

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2011-11 Skew-symmetric and symmetric matrices

Prove that for every skew-symmetric matrix A, there are symmetric matrices B and C such that A=BC-CB.

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