# 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: 2012-12 Big partial sum

Let A be a finite set of complex numbers. Prove that there exists a subset B of A such that $\bigl\lvert\sum_{z\in B} z\bigr\lvert \ge \frac{ 1}{\pi}\sum_{z\in A} \lvert z\rvert.$

The best solution was submitted by Minjae Park (박민재), 2011학번. Congratulations!

Here is Solution of Problem 2012-12.

Two incorrect solutions were submitted (M.J.L., W.S.J.).

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# Solution: 2012-11 Dividing a circle

Let f be a continuous function from [0,1] such that f([0,1]) is a circle. Prove that there exists two closed intervals $$I_1, I_2 \subseteq [0,1]$$ such that $$I_1\cap I_2$$ has at most one point, $$f(I_1)$$ and $$f(I_2)$$ are semicircles, and $$f(I_1)\cup f(I_2)$$ is a circle.

The best solution was submitted by Myeongjae Lee (이명재), 2012학번. Congratulations!

Here is his Solution of Problem 2012-11.

Alternative solution was submitted by 박민재(2011학번, +3). Two incorrect solutions were submitted (W.S.J., K.M.P.).

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# 2012-13 functions for an inequality

Determine all nonnegative functions f(x,y) and g(x,y) such that $\left(\sum_{i=1}^n a_i b_i \right)^2 \le \left( \sum_{i=1}^n f(a_i,b_i)\right) \left(\sum_{i=1}^n g(a_i,b_i)\right) \le \left(\sum_{i=1}^n a_i^2\right) \left(\sum_{i=1}^n b_i^2\right)$ for all reals $$a_i$$, $$b_i$$ and all positive integers n.

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# 2012-12 Big partial sum

Let A be a finite set of complex numbers. Prove that there exists a subset B of A such that $\bigl\lvert\sum_{z\in B} z\bigr\lvert \ge \frac{ 1}{\pi}\sum_{z\in A} \lvert z\rvert.$

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