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.

Let \(A\) be a \(2\times 2\) matrix. Prove that if \(Av_1=\lambda_1v_1\) and \(Av_2=\lambda_2v_2\) for distinct reals \(\lambda_1\) and \(\lambda_2\) and nonzero vectors \(v_1\) and \(v_2\), then both columns of \(A-\lambda_1 I\) is a multiple of \(v_2\).

The best solution was submitted by Lee, Jongwon (이종원, 수리과학과 2014학번). Congratulations!

Here is his solution of problem 2018-12.

Alternative solutions were submitted by 고성훈 (2018학번, +3), 권홍 (중앙대 물리학과, +3), 길현준 (2018학번, +3), 김태균 (수리과학과 2016학번, +3), 이본우 (수리과학과 2017학번, +3), 채지석 (수리과학과 2016학번, +3), 한준호 (수리과학과 2015학번, +3).

On a math exam, there was a question that asked for the largest angle of the triangle with sidelengths \(21\), \(41\), and \(50\). A student obtained the correct answer as follows:

Let \(x\) be the largest angle. Then,
\[
\sin x = \frac{50}{41} = 1 + \frac{9}{41}.
\]
Since \( \sin 90^{\circ} = 1 \) and \( \sin 12^{\circ} 40′ 49” = 9/41 \), the angle \( x = 90^{\circ} + 12^{\circ} 40′ 49” = 102^{\circ} 40′ 49”\).

Find the triangle with the smallest area with integer sidelengths and possessing this property (that the wrong argument as above gives the correct answer).

The best solution was submitted by Han, Junho (한준호, 수리과학과 2015학번). Congratulations!

Here is his solution of problem 2018-11.

Alternative solutions were submitted by 이종원 (수리과학과 2014학번, +3),
채지석 (수리과학과 2016학번, +3), 고성훈 (2018학번, +2), 이본우 (수리과학과 2017학번, +2). One incorrect solution was submitted.

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).

For a positive integer \( n \), let \( S(n) \) be the sum of all decimal digits in \( n \), i.e., if \( n = n_1 n_2 \dots n_m \) is the decimal expansion of \( n \), then \( S(n) = n_1 + n_2 + \dots + n_m \). Find all positive integers \( n \) and \( r \) such that \( (S(n))^r = S(n^r) \).

The best solution was submitted by Lee, Jongwon (이종원, 수리과학과 2014학번). Congratulations!

Here is his solution of problem 2018-09.

Alternative solutions were submitted by 채지석 (수리과학과 2016학번, +3), 한준호 (수리과학과 2015학번, +3), 이본우 (수리과학과 2017학번, +3), 권홍 (중앙대 물리학과, +2).

Let \(a_1\), \(a_2\), \(\ldots\), \(a_m\) be distinct positive integers. Prove that if \(m>2\sqrt{N}\), then there exist \(i\), \(j\) such that the least common multiple of \(a_i\) and \(a_j\) is greater than \(N\).

The best solution was submitted by Bonwoo Lee (이본우, 수리과학과 2017학번). Congratulations!

Here is his solution of problem 2018-08.

Alternative solutions were submitted by 이종원 (수리과학과 2014학번, +3), 김태균 (수리과학과 2016학번, +3), 한준호 (수리과학과 2015학번, +3), 이재우 (함양고등학교 3학년, +3).

Let \( S \) be an \( (n+1) \times (n+1) \) matrix defined by
\[
S_{ij} = \begin{cases}
(n+1)-i & \text{ if } j=i+1, \\
i-1 & \text{ if } j=i-1, \\
0 & \text{ otherwise. }
\end{cases}
\]
Find all eigenvalues of \( S \).

The best solution was submitted by Lee, Jongwon (이종원, 수리과학과 2014학번). Congratulations!

Here is his solution of problem 2018-07.

Alternative solutions were submitted by 한준호 (수리과학과 2015학번, +3), 채지석 (수리과학과 2016학번, +3), Hitesh Kumar (Imperial College London, +2), 고성훈 (2018학번, +2).