Compute the following integral \[ \int_{0}^{\pi/2} \log{ (2 \cos{x} )} dx \].

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Compute the following integral \[ \int_{0}^{\pi/2} \log{ (2 \cos{x} )} dx \].

Let \( p_n \) be the \(n\)-th prime number, \( p_1 = 2, p_2 = 3, p_3 = 5, \dots \). Prove that the following series converges:

\[

\sum_{n=1}^{\infty} \frac{1}{p_n} \prod_{k=1}^n \frac{p_k -1}{p_k}.

\]

The best solution was submitted by 김기현 (수리과학과 대학원생). Congratulations!

Here is his solution of problem 2019-05.

Other solutions were submitted by 강한필 (전산학부 2016학번), 고성훈 (2018학번, +3), 길현준 (2018학번, +3), 김기수 (수리과학과 2018학번), 김기현 (수리과학과 대학원생), 김태균 (수리과학과 2016학번), 박항 (전산학부 2013학번), 신원석 (서울대학교 컴퓨터공학부), 이본우 (수리과학과 2017학번, +3), 이정환 (수리과학과 2015학번, +3), 조재형 (수리과학과 2016학번, +3), 채지석 (수리과학과 2016학번), 최백규 (생명과학과 2016학번, +3), 김민서 (2019학번, +2), 윤창기 (서울대학교 화학과).

Let \( p_n \) be the \(n\)-th prime number, \( p_1 = 2, p_2 = 3, p_3 = 5, \dots \). Prove that the following series converges:

\[

\sum_{n=1}^{\infty} \frac{1}{p_n} \prod_{k=1}^n \frac{p_k -1}{p_k}.

\]

Ten mathematicians sit at a round table. Each has a certain amount of food. At each full minute, every mathematician divides his share of food into two equal parts and hands it out to the two people seated closest to him in counter-clockwise direction. How will the food be distributed at the end of a long evening? Does the answer change if instead every mathematician shares his food with the two people sitting immediately next to him?

The best solution was submitted by 채지석 (수리과학과 2016학번). Congratulations!

Here is his solution of problem 2019-04.

Other solutions were submitted by 고성훈 (2018학번, +3), 길현준 (2018학번, +3), 김기수 (수리과학과 2018학번), 김기현 (수리과학과 대학원생), 김민서 (2019학번, +3), 김태균 (수리과학과 2016학번), 이본우 (수리과학과 2017학번, +3), 이원영 (2019학번), 이정환 (수리과학과 2015학번, +3), 이종서 (2019학번), 조재형 (수리과학과 2016학번, +3), 최백규 (생명과학과 2016학번, +3). Late solutions are not graded.