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

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

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).

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\[ \sum_{\pi \in S_n } \frac{1}{x_{\pi(1)}} \frac{1}{x_{\pi(1)}+x_{\pi(2)}}\cdots \frac{1}{x_{\pi(1)}+\cdots+ x_{\pi(n-1)}}. \]

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).

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

Our first problem for Spring 2012 semester will be posted on Feb. 10, 3pm.