# Concluding 2012 Fall

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

• 1st prize: Lee, Myeongjae  (이명재) – 2012학번
• 2nd prize: Kim, Taeho (김태호) – 수리과학과 2011학번
• 3rd prize: Park, Minjae (박민재) – 2011학번
• 4th prize: Suh, Gee Won (서기원) – 수리과학과 2009학번
• 5th prize: Lim, Hyunjin (임현진) – 물리학과 2010학번

Congratulations! We again have very good prizes this semester – iPad 16GB for the 1st prize, iPad Mini 16GB for the 2nd prize, etc. 이명재 (2012학번) 32
김태호 (2011학번) 30
박민재 (2011학번) 25
서기원 (2009학번) 21
임현진 (2010학번) 17
김주완 (2010학번) 10
조상흠 (2010학번) 8
임정환 (2009학번) 7
김홍규 (2011학번) 5
곽걸담 (2011학번) 5
김지원 (2010학번) 5
이신영 (2012학번) 5
윤영수 (2011학번) 5
엄태현 (2012학번) 4
조준영 (2012학번) 3
박종호 (2009학번) 3
정종헌 (2012학번) 2
장영재 (2011학번) 2
양지훈 (2010학번) 2
최원준 (2009학번) 2
김지홍 (2007학번) 2
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# Solution: 2012-24 Determinant of a Huge Matrix

Consider all non-empty subsets $$S_1,S_2,\ldots,S_{2^n-1}$$ of $$\{1,2,3,\ldots,n\}$$. Let $$A=(a_{ij})$$ be a $$(2^n-1)\times(2^n-1)$$ matrix such that $a_{ij}=\begin{cases}1 & \text{if }S_i\cap S_j\ne \emptyset,\\0&\text{otherwise.}\end{cases}$ What is $$\lvert\det A\rvert$$?

The best solution was submitted by Kim, Taeho (김태호), 수리과학과 2011학번. Congratulations!

Here is his Solution of Problem 2012-24.

Alternative solutions were submitted by 이명재 (2012학번, +3), 임현진 (물리학과 2010학번, +3), 정종헌 (2012학번, +2),  어수강 (서울대학교 수리과학부 석사과정, +3).

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# Solution: 2012-15 Functional Equation

Let $$n$$ be a fixed positive integer. Find all functions $$f:\mathbb{R}\to\mathbb{R}$$ satisfying $f(x^{n+1}-y^{n+1})=(x-y)[f(x)^n+f(x)^{n-1}f(y)+\cdots+f(x)f(y)^{n-1}+f(y)^n].$

The best solution was submitted by Kim, Taeho (김태호), 수리과학과 2011학번. Congratulations!

Here is his Solution of Problem 2012-15.

Alternative solutions were submitted by 임정환 (수리과학과 2009학번, +3), 곽걸담 (물리학과 2011학번, +2), 서기원 (수리과학과 2009학번, +2),  김홍규 (수리과학과 2011학번, +2), 김지원 (수리과학과 2010학번, +2), 이명재 (2012학번, +2), 조상흠 (수리과학과 2010학번, +2). There were 2 incorrect submissions (LHJ, KDR).

GD Star Rating # Concluding 2012 Spring

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

• 1st prize: Park, Minjae (박민재) – 2011학번
• 2nd prize: Lee, Myeongjae  (이명재) – 2012학번
• 3rd prize: Suh, Gee Won (서기원) – 수리과학과 2009학번
• 4th prize: Cho, Junyoung (조준영) – 2012학번
• 5th prize: Kim, Taeho (김태호) – 수리과학과 2011학번

Congratulations! As announced earlier, we have nicer prize this semester – iPad 16GB for the 1st prize, iPod Touch 32GB for the 2nd prize, etc. 박민재 (2011학번) 41
이명재 (2012학번) 34
서기원 (2009학번) 29
조준영 (2012학번) 17
김태호 (2011학번) 16
서동휘 (2009학번) 5
임정환 (2009학번) 5
이영훈 (2011학번) 4
임창준 (2012학번) 3
Phan Kieu My (2009학번) 3
장성우 (2010학번) 2
홍승한 (2012학번) 2
윤영수 (2011학번) 2
변성철 (2011학번) 2
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# Solution of 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.

The best solution was submitted by Taeho Kim (김태호, 수리과학과 2011학번). Congratulations!

Here is his Solution of Problem 2012-7.

Alternative solutions were submitted by 서기원 (수리과학과 2009학번, +3), 박민재 (2011학번, +3), 조준영 (2012학번, +3), 이명재 (2012학번, +3), 정우석 (서강대 2011학번, +3), 천용 (전남대 의예과 2011학번 +3), 어수강 (서울대학교 석사과정, +2).

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# Solution: 2011-17 Infinitely many solutions

Let f(n) be the maximum positive integer m such that the sum of all positive divisors of m is less than or equal to n. Find all positive integers k such that there are infinitely many positive integers n satisfying the equation n-f(n)=k.

The best solution was submitted by Taeho Kim (김태호), 2011학번. Congratulations!

Here is his Solution of Problem 2011-17.

Alternative solutions were submitted by 김범수 (수리과학과 2010학번, +3), 서기원 (수리과학과 2009학번, +3), 박승균 (수리과학과 2008학번, +3), 장경석 (2011학번, +3), 구도완 (해운대고등학교 3학년, +3), 손동현 (유성고등학교 2학년, +2), 어수강 (서울대학교 석사과정, +2).

Update: I forgot to add 최민수 (2011학번, +3) into the list of people submitted alternative solutions.

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