For given positive real numbers \(a_1,\ldots,a_k\) and for each integer n≥k, let \(a_{n+1}\) be the geometric mean of \( a_n, a_{n-1}, a_{n-2}, \ldots, a_{n-k+1}\). Prove that \( \lim_{n\to\infty} a_n\) exists and compute this limit.

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

Here is his Solution of Problem 2012-5.

Alternative solutions were submitted by 박민재 (2011학번, +3, Solution), 김태호 (2011학번, +3, Solution), 이명재 (2012학번, +3), 박훈민 (대전과학고등학교 2학년, +3), 윤영수 (2011학번, +2), 조준영 (2012학번, +2), 변성철 (2011학번, +2), 정우석 (서강대학교 자연과학부 2011학번, +2). One incorrect solution was received.

Find the smallest and the second smallest odd integers n satisfying the following property: \[ n=x_1^2+y_1^2 \text{ and } n^2=x_2^2+y_2^2 \] for some positive integers \(x_1,y_1,x_2,y_2\) such that \(x_1-y_1=x_2-y_2\).

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

Here is his Solution of Problem 2012-4.

Alternative solutions were submitted by 조준영 (2012학번, +3), 서기원 (수리과학과 2009학번, +3), 임창준 (2012학번, +3), 홍승한 (2012학번, +2), 이명재 (2012학번, +2), 김현수 (?, +3), 천용 (전남대, +2). One incorrect solution was received.

Compute \[ f(x)= \int_{0}^1 \frac{\log (1- 2t\cos x + t^2) }{t} dt.\]

The best solution was submitted by Younghun Lee (이영훈), 2011학번.

Here is his Solution of Problem 2012-3.

Alternative solutions were submitted by 조준영 (2012학번, +3, Solution), 김태호 (2011학번, +3), 서기원 (수리과학과 2009학번, +3), 박민재 (2011학번, +3, Solution), 이명재 (2012학번, +3), 서동휘 (수리과학과 2009학번, +2), 임정환 (수리과학과 2009학번, +2), 김현수 (?, +2), 정우석 (서강대 자연과학부 2011학번, +2), 조위지 (Stanford Univ. 물리학과 박사과정, +3, Solution).

Let n be a positive integer and let S_n be the set of all permutations on {1,2,…,n}. Assume \( x_1+x_2 +\cdots +x_n =0\) and \(\sum_{i\in A} x_i\neq 0 \) for all nonempty proper subsets A of {1,2,…,n}. Find all possible values of\[ \sum_{\pi \in S_n } \frac{1}{x_{\pi(1)}} \frac{1}{x_{\pi(1)}+x_{\pi(2)}}\cdots \frac{1}{x_{\pi(1)}+\cdots+ x_{\pi(n-1)}}. \]

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

Here is his Solution of Problem 2012-2.

Alternative solutions were submitted by 이명재 (2012학번, +3,  Solution), 조준영 (2012학번, +3), 김태호 (2011학번, +3), 박민재 (2011학번, +3, Solution), 서동휘 (수리과학과 2009학번, +3), 임정환 (수리과학과 2009학번, +3), 박훈민 (대전과학고 1학년, +3, Solution), 조위지 (Stanford Univ. 물리학과 박사과정, +3, Solution), 김건형 (서울대 컴퓨터공학과 2012학번, +3).

Compute tan^-1(1) -tan^-1(1/3) + tan^-1(1/5) – tan^-1(1/7) + … .

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

Here is his Solution of Problem 2012-1.

Alternative solutions were submitted by 서기원 (수리과학과 2009학번, +3), 조준영 (2012학번, +3), 조위지 (Stanford Univ. 물리학과 박사과정, +3, Solution), 박훈민 (대전과학고 1학년, +3), 이명재 (2012학번, +2), 장성우 (2010학번, +2).

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).

Find all n≥2 such that the polynomial xⁿ-x^n-1-x^n-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.