# Solution: 2014-01 Uniform convergence

Let $$f$$ be a real-valued continuous function on $$[ 0, 1]$$. For a positive integer $$n$$, define
$B_n(f; x) = \sum_{j=0}^n f( \frac{j}{n}) {n \choose j} x^j (1-x)^{n-j}.$
Prove that $$B_n (f; x)$$ converges to $$f$$ uniformly on $$[0, 1 ]$$ as $$n \to \infty$$.

The best solution was submitted by 김범수. Congratulations!

Similar solutions are submitted by 권현우(+3), 박경호(+3), 오동우(+3), 이시우(+3), 이종원(+3), 이주호(+3), 장경석(+3), 장기정(+3), 정성진(+3), 정진야(+3), 조준영(+3), 채석주(+3), 한대진(+3), 황성호(+3). Thank you for your participation.

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# 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: 2013-04 Largest eigenvalue of a symmetric matrix

Let $$H$$ be an $$N \times N$$ real symmetric matrix. Suppose that $$|H_{kk}| < 1$$ for $$1 \leq k \leq N$$. Prove that, if $$|H_{ij}| > 4$$ for some $$i, j$$, then the largest eigenvalue of $$H$$ is larger than $$3$$.

The best solution was submitted by 김범수, 10학번. Congratulations!

Other solutions were submitted by 라준현(08학번, +3), 서기원(09학번, +3), 김호진(09학번, +3), 김범수(10학번, +3), 박훈민(13학번, +3), 노수현(13학번, +2). Thank you for your participation.

GD Star Rating # 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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For a nonnegative integer n, let $$F_n(x)=\sum_{m=0}^n \frac{(-2)^m (2n-m)! \Gamma(x+1)}{m! (n-m)! \Gamma(x-m+1)}$$. Find all x such that Fn(x)=0.