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.

Let f(x) be a continuous function on I=[a,b], and let g(x) be a differentiable function on I. Let g(a)=0 and c≠0 a constant. Prove that if

|g(x) f(x)+c g′(x)|≤|g(x)| for all x∈I,

then g(x)=0 for all x∈I.

The best solution was submitted by Seungkyun Park (박승균), 수리과학과 2008학번. Congratulations!

Here is his Solution of Problem 2011-18.

Alternative solutions were submitted by 김범수 (수리과학과 2010학번, +3), 장경석 (2011학번, +3),  김태호 (2011학번, +2), 김재훈 (EEWS대학원, +2).

Let f(n) be the maximum positive integer m such that the sum of all positive divisors of m is less than or equal to n. Find all positive integers k such that there are infinitely many positive integers n satisfying the equation n-f(n)=k.

The best solution was submitted by Taeho Kim (김태호), 2011학번. Congratulations!

Here is his Solution of Problem 2011-17.

Alternative solutions were submitted by 김범수 (수리과학과 2010학번, +3), 서기원 (수리과학과 2009학번, +3), 박승균 (수리과학과 2008학번, +3), 장경석 (2011학번, +3), 구도완 (해운대고등학교 3학년, +3), 손동현 (유성고등학교 2학년, +2), 어수강 (서울대학교 석사과정, +2).

Update: I forgot to add 최민수 (2011학번, +3) into the list of people submitted alternative solutions.

Let A₁, A₂, A₃, …, A_n be finite sets such that |A_i| is odd for all 1≤i≤n and |A_i∩A_j| is even for all 1≤i<j≤n. Prove that it is possible to pick one element a_i in each set A_i so that a₁, a₂, …,a_n are distinct.

The best solution was submitted by Ilhee Kim (김일희) and Ringi Kim (김린기), Graduate Students, Princeton University.

Here is Solution of Problem 2011-16.

Alternative solutions were submitted by 강동엽 (전산학과 2009학번, +3, solution by 강동엽), 문상혁 & 박상현 (2010학번, +3), 장경석 (2011학번, +3), 이재석 (수리과학과 2007학번, +2). Three incorrect solutions were submitted by B. Kim, J. Lee, Y. Park.