# Concluding Spring 2013

The top 5 participants of the semester are:

• 1st: 라준현 (08학번): 38 points
• 2nd: 서기원 (09학번): 29 points
• T-3rd: 김호진 (09학번): 25 points
• T-3rd: 황성호 (13학번): 25 points
• 5th: 김범수 (10학번): 19 points

Hearty congratulations to the prize winners! The prize ceremony will be held on Jun. 19 (Wed.) at 2PM.

We thank all of the participants for the nice solutions and your intereset you showed for POW. We hope to see you next semester with even better problems.

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# Solution: 2013-12 Equilateral triangle in R^n

Let $$A = \{ (a_1, a_2, \cdots, a_n : a_i = \pm 1 \, (i = 1, 2, \cdots, n) \} \subset \mathbb{R}^n$$. Prove that, for any $$X \subset A$$ with $$|X| > 2^{n+1}/n$$, there exist three distinct points in $$X$$ that are the vertices of an equilateral triangle.

The best solution was submitted by 서기원, 09학번. Congratulations!

Similar solutions were submitted by 라준현(08학번, +3), 김호진(09학번, +3), 황성호(13학번, +3), 박정현(일반, +3), 정요한(서울시립대, +3). Thank you for your participation.

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# Solution: 2013-08 Minimum of a set involving polynomials with integer coefficients

Let $$p$$ be a prime number. Let $$S_p$$ be the set of all positive integers $$n$$ satisfying
$x^n – 1 = (x^p – x + 1) f(x) + p g(x)$
for some polynomials $$f$$ and $$g$$ with integer coefficients. Find all $$p$$ for which $$p^p -1$$ is the minimum of $$S_p$$.

The best solution was submitted by 서기원, 09학번. Congratulations!

Other solutions were submitted by 라준현(08학번, +3), 어수강(서울대, +3). Thank you for your participation.

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# 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: 2012-17 Two Tangent Functions in a Series

Let $$m$$ and $$n$$ be odd integers. Determine $\sum_{k=1}^\infty \frac{1}{k^2}\tan\frac{k\pi}{m}\tan \frac{k\pi}{n}.$

The best solution was submitted by Suh, Gee Won (서기원), 수리과학과 2009학번. Congratulations!

Here is his Solution of Problem 2012-17.

One incorrect solution was received (PHM).

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# 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: 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: 2012-2 sum with a permutation

Let n be a positive integer and let Sn be the set of all permutations on {1,2,…,n}. Assume $$x_1+x_2 +\cdots +x_n =0$$ and $$\sum_{i\in A} x_i\neq 0$$ for all nonempty proper subsets A of {1,2,…,n}. Find all possible values of$\sum_{\pi \in S_n } \frac{1}{x_{\pi(1)}} \frac{1}{x_{\pi(1)}+x_{\pi(2)}}\cdots \frac{1}{x_{\pi(1)}+\cdots+ x_{\pi(n-1)}}.$

The best solution was submitted by Gee Won Suh (서기원), 수리과학과 2009학번. Congratulations!

Here is his Solution of Problem 2012-2.

Alternative solutions were submitted by 이명재 (2012학번, +3,  Solution), 조준영 (2012학번, +3), 김태호 (2011학번, +3), 박민재 (2011학번, +3, Solution), 서동휘 (수리과학과 2009학번, +3), 임정환 (수리과학과 2009학번, +3), 박훈민 (대전과학고 1학년, +3, Solution), 조위지 (Stanford Univ. 물리학과 박사과정, +3, Solution), 김건형 (서울대 컴퓨터공학과 2012학번, +3).

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# Concluding 2011 Fall

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

1st prize: Jang, Kyoungseok (장경석) – 2011학번

2nd prize: Suh, Gee Won (서기원) – 수리과학과 2009학번

3rd prize: Kim, Bumsu (김범수) – 수리과학과 2010학번

4th prize: Park, Seungkyun (박승균) – 수리과학과 2008학번

5th prize: Park, Minjae (박민재) – 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.

장경석 (2011학번) 28 pts
서기원 (2009학번) 27 pts
김범수 (2010학번) 22 pts
박승균 (2008학번) 14 pts
박민재 (2011학번) 13 pts
강동엽 (2009학번) 11 pts
김태호 (2011학번) 9 pts
김원중 (2011학번) 3 pts
곽영진 (2011학번) 3 pts
조상흠 (2010학번) 3 pts
라준현 (2008학번) 3 pts
배다슬 (2008학번) 3 pts
이재석 (2007학번) 3 pts
최민수 (2011학번) 3 pts
문상혁 (2010학번) 2 pts
박상현 (2010학번) 2 pts

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# Solution: 2011-20 Double infinite series

For a real number x, let d(x)=minn:integer (x-n)2. Evaluate the following double infinite series:
. . . + 8 d(x/8)+4 d(x/4) + 2 d(x/2) + d(x)  + d(2x) / 2 + d(4x)/4 + d(8x)/8 + . . .

The best solution was submitted by Gee Won Suh (서기원), 수리과학과 2009학번. Congratulations!

Here is his Solution of Problem 2011-20.

Alternative solutions were submitted by 박승균 (수리과학과 2008학번, Alternative Solution, +3) and 장경석 (2011학번, +3).

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