# 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 $\sum_{a_1=0}^{t_1} \sum_{a_2=0}^{t_2}\cdots \sum_{a_n=0}^{t_n} (-1)^{t_1+t_2+\cdots+t_n+a_1+a_2+\cdots+a_n}\left( \prod_{i=1}^n \binom{t_i}{a_i} \right)p(a_1,a_2,\ldots,a_n).$

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# 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.

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# Solution: 2011-8 Geometric Mean

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.)

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# 2011-8 Geometric Mean

Let f be a continuous function on [0,1]. Prove that $\lim_{n\to \infty}\int_0^1 \cdots \int_0^1 f(\sqrt[n]{x_1 x_2 \cdots x_n } ) dx_1 dx_2 \cdots dx_n = f(1/e).$

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Let f(n) be the largest integer k such that n! is divisible by $$n^k$$. Prove that $\lim_{n\to \infty} \frac{(\log n)\cdot \max_{2\le i\le n} f(i)}{n \log\log n}=1.$