Evaluate the sum \[ \sum_{k=0}^{[n/2]} (-4)^{n-k} \binom{n-k}{k} ,\] where [x] denotes the greatest integer less than or equal to x.

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

Here is his Solution of Problem 2011-24.

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

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 a_n+i=a_n with a_n+i=a_i.)

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).

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 F_n(x)=0.

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

Here is his Solution of Problem 2011-21.

For a real number x, let d(x)=min_n: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).