Monthly Archives: May 2012

KAIST Math Problem of the Week 2012 봄 시상식 박민재 이명재 서기원 조준영 김태호

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.

KAIST Math Problem of the Week 2012 봄 시상식 박민재 이명재 서기원 조준영 김태호

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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