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).

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.

Suppose that \( T \) is an \( N \times N \) matrix
\[
T = \begin{pmatrix}
a_1 & b_1 & 0 & \cdots & 0 \\
b_1 & a_2 & b_2 & \ddots & \vdots \\
0 & b_2 & a_3 & \ddots & 0 \\
\vdots & \ddots & \ddots & \ddots & b_{N-1} \\
0 & \cdots & 0 & b_{N-1} & a_N
\end{pmatrix}
\]
with \( b_i > 0 \) for \( i =1, 2, \dots, N-1 \). Prove that \( T \) has \( N \) distinct eigenvalues.

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

Here is his solution of problem 2019-03.

Other solutions were submitted by 고성훈 (2018학번, +3), 길현준 (2018학번, +3), 김기수 (수리과학과 2018학번), 김민서 (2019학번, +3), 박현영 (전기및전자공학부 2016학번, +3), 이본우 (수리과학과 2017학번, +3), 이상윤 (UCLA, +3), 이정환 (수리과학과 2015학번, +3), 조재형 (수리과학과 2016학번, +3), 최백규 (생명과학과 2016학번, +3). Late solutions are not graded.

For any positive integers m and n, show that

\[ C_{n,m} = \frac{(mn)!}{(m!)^n n!} \] is an integer.

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

Here is his solution of problem 2019-02.

Other solutions were submitted by Alfonso Alvarenga (전산학부 2015학번, +3), 고성훈 (2018학번, +3), 길현준 (2018학번, +3), 김기수 (수리과학과 2018학번), 김민서 (2019학번, +3), 김태균 (수리과학과 2016학번, +3), 박건규 (수리과학과 2015학번, +3), 박수찬 (전산학부 2017학번, +3), 박현영 (전기및전자공학부 2016학번, +3), 윤현민 (수리과학과 2018학번), 이본우 (수리과학과 2017학번, +3), 이상윤 (UCLA, +3), 이정환 (수리과학과 2015학번, +3), 이종서 (2019학번, +3), 이태영 (수리과학과 졸업생, +3), 조재형 (수리과학과 2016학번, +3), 조정휘 (건국대학교 수학과 2014학번, +3), 채지석 (수리과학과 2016학번, +3), 최백규 (생명과학과 2016학번, +3). Late solutions are not graded.

Suppose that \( \Pi \) is a closed polygon in the plane. If \( \Pi \) is equilateral \( k \)-gon, and if \( A \) is the area of \( \Pi \), and \( L \) the length of its boundary, prove that
\[
\frac{A}{L^2} \leq \frac{1}{4k} \cot \frac{\pi}{k} \leq \frac{1}{4\pi}.
\]

The best solution was submitted by 윤창기 (서울대학교 화학과). Congratulations!

Here is his solution of problem 2019-01.

Similar solutions were submitted by 길현준 (2018학번, +3), 조재형 (수리과학과 2016학번, +3), 김태균 (수리과학과 2016학번, +2). Alternative solution was submitted by 고성훈 (2018학번, +3). Four incorrect solutions were received.