Find all integers \( n \) such that the following holds:

There exists a set of \( 2n \) consecutive squares \( S = \{ (m+1)^2, (m+2)^2, \dots, (m+2n)^2 \} \) (\( m \) is a nonnegative integer) such that \( S = A \cup B \) for some \( A \) and \( B \) with \( |A| = |B| = n \) and the sum of elements in \( A \) is equal to the sum of elements in \( B \).

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

Here is his solution of problem 2019-19.

An incorrect solution was submitted.

Let \( A, B \) be \( n \times n \) Hermitian matrices. Find all positive integer \( n \) such that the following statement holds:

“If \( AB – BA \) is singular, then \( A \) and \( B \) have a common eigenvector.”

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

Here is his solution of problem 2019-14.

A similar solution was submitted by 하석민 (수리과학과 2017학번, +3). Late solutions are not graded.

Suppose that \( f: \mathbb{R} \to \mathbb{R} \) is differentiable and \( \max_{ x \in \mathbb{R}} |f(x)| = M < \infty \). Prove that \[ \int_{-\infty}^{\infty} (|f'|^2 + |f|^2) \geq 2M^2. \]

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

Here is his solution of problem 2019-07.

Other solutions were submitted by 고성훈 (2018학번, +3), 길현준 (2018학번, +3), 김기택 (수리과학과 2015학번, +3), 김민서 (2019학번, +3), 김태균 (수리과학과 2016학번, +3), 박재원 (2019학번, +3), 오윤석 (2019학번, +3), 윤영환 (한양대학교, +3), 이본우 (수리과학과 2017학번, +3), 이원용 (2019학번, +3), 이정환 (수리과학과 2015학번, +3), 정의현 (수리과학과 대학원생, +3), 최백규 (생명과학과 2016학번, +3).

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.

Does there exist a (possibly \(n\)-dependent) constant \( C \) such that
\[
\frac{C}{a_n} \sum_{1 \leq i < j \leq n} (a_i-a_j)^2 \leq \frac{a_1+ \dots + a_n}{n} – \sqrt[n]{a_1 \dots a_n} \leq \frac{C}{a_1} \sum_{1 \leq i < j \leq n} (a_i-a_j)^2
\]
for any \( 0 < a_1 \leq a_2 \leq \dots \leq a_n \)?

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

Here is his solution of problem 2018-21.

Alternative solutions were submitted by 하석민 (수리과학과 2017학번, +3),
이본우 (수리과학과 2017학번, +2). One incorrect submission was received.

Let \( n \) be a positive integer. Suppose that \( a_1, a_2, \dots, a_n \) are non-zero integers and \( b_1, b_2, \dots, b_n\) are positive integers such that \( (b_i, b_n) = 1 \) for \( i = 1, 2, \dots, n-1 \). Prove that the Diophantine equation
\[
a_1 x_1^{b_1} + a_2 x_2^{b_2} + \dots + a_n x_n^{b_n} = 0
\]
has infinitely many integer solutions \( (x_1, x_2, \dots, x_n) \).

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

Here is his solution of problem 2018-15.

An alternative solution was submitted by Saba Dzmanashvili (수리과학과 2017학번, +3), 강한필 (전산학부 2016학번, +3), 권홍 (중앙대 물리학과, +3), 이본우 (수리과학과 2017학번, +3), 이재우 (함양고등학교 3학년, +3), 최백규 (생명과학과 2016학번, +3), 길현준 (2018학번, +2), 김태균 (수리과학과 2016학번, +2).

Assume that \( x \in \mathbb{R}^n \) with at least \( k \) non-zero entries \( ( k> 0 ) \). Let
\[
A = \{ y \in \{-1, 1\}^n : y \cdot x = 0 \}.
\]
Prove that \( |A| \leq k^{-1/2} 2^n \).

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

Here is his solution of problem 2018-13.

An alternative solution was submitted by 이대석 (수리과학과 2017학번, +3). Two incorrect solutions were received.

Given a stick of length 1, we choose two points at random and break it into three pieces. Compute the probability that these three pieces form an acute triangle.

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

Here is his solution of problem 2018-10.

Alternative solutions were submitted by 고성훈 (2018학번, +3), 이종원 (수리과학과 2014학번, +3), 한준호 (수리과학과 2015학번, +3), 김태균 (수리과학과 2016학번, +2), 이본우 (수리과학과 2017학번, +2), 이준성 (상문고등학교 2학년, +2), Harrison Zhu (Imperial College London, +2).