# Surprise: More and Better Prizes this semester

The prize for KAIST Math POW is getting better, thanks to the department support. From this Fall semester, we will have the following as the prizes:

2nd prize: iPod Touch 32GB

3rd prize: 5 WEEKDAY DINNER gift certificates for  for a buffet restaurant in Yuseong

4th prize: 3 WEEKDAY DINNER gift certificates for a buffet restaurant in Yuseong

5th prize: 2 WEEKDAY DINNER gift certificates for a buffet restaurant in Yuseong

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# Solution: 2011-22 Seoul Subway Line 2

In Seoul Subway Line 2,  subway stations are placed around a circular subway line. Assume that each segment of Seoul Subway Line 2 has a fixed price. Suppose that you hid money at each subway station so that the sum of the money is only enough for one roundtrip around Seoul Subway Line 2.

Prove that there is a station that you can start and take a roundtrip tour of Seoul Subway Line 2 while paying each segment by the money collected at visited stations.

The best solution was submitted by Kang, Dongyub (강동엽), 전산학과 2009학번. Congratulations!

Here is his Solution of Problem 2011-22. (typo in the lemma: replace an+i=an with an+i=ai.)

Alternative solutions were submitted by 서기원 (수리과학과 2009학번, +3 Alternative Solution), 장경석 (2011학번, +3), 김태호 (2011학번, +3), 김범수 (수리과학과 2010학번, +3), 박준하 (하나고등학교 2학년, +3).

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# 2011-23 Constant Function

Let $$f:\mathbb{R}^n\to \mathbb{R}^{n-1}$$ be a function such that for each point a in $$\mathbb{R}^n$$, the limit $$\lim_{x\to a} \frac{|f(x)-f(a)|}{|x-a|}$$ exists. Prove that f is a constant function.

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# Solution: 2011-21 Zeros

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.

The best solution was submitted by Bumsu Kim (김범수), 수리과학과 2010학번.

Here is his Solution of Problem 2011-21.

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# 2011-22 Seoul Subway Line 2

In Seoul Subway Line 2,  subway stations are placed around a circular subway line. Assume that each segment of Seoul Subway Line 2 has a fixed price. Suppose that you hid money at each subway station so that the sum of the money is only enough for one roundtrip around Seoul Subway Line 2.

Prove that there is a station that you can start and take a roundtrip tour of Seoul Subway Line 2 while paying each segment by the money collected at visited stations.

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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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# 2011-21 Zeros

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.

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# Solution: 2011-19 Irreducible polynomial

Find all n≥2 such that the polynomial xn-xn-1-xn-2-…-x-1 is irreducible over the rationals.

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

Here is his Solution of Problem 2011-19.

One incorrect solution by W.J. Kim was submitted.

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