# 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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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.