# Solution: 2011-23 Constant Function

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.

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# 2012-7 Product of Sine

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.

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# Good luck for your midterm exam

Good luck next week for your midterm exam! We take break and return on March 30, Friday.

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# Solution: 2012-6 Matrix modulo p

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).

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# 2012-6 Matrix modulo p

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.

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# Solution: 2012-5 Iterative geometric mean

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.

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# Solution: 2012-4 Sum of squares

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.

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# 2012-5 Iterative geometric mean

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.

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# 2012-4 Sum of squares

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 $$x_1-y_1=x_2-y_2$$.

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Compute $f(x)= \int_{0}^1 \frac{\log (1- 2t\cos x + t^2) }{t} dt.$