# 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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Prove that for each positive integer $$n$$, there exist $$n$$ real numbers $$x_1,x_2,\ldots,x_n$$ such that $\sum_{j=1}^n \frac{x_j}{1-4(i-j)^2}=1 \text{ for all }i=1,2,\ldots,n$ and $\sum_{j=1}^n x_j=\binom{n+1}{2}.$