Concluding 2018 Spring

Thanks all for participating POW actively. Here’s the list of winners:

1st prize (Gold): Lee, Jongwon (이종원, 수리과학과 2014학번)
2nd prize (Silver): Chae, Jiseok (채지석, 수리과학 과 2016학번)
2nd prize (Silver): Han, Joon Ho (한준호,수리과학과 2015학번)
2nd prize (Silver): Lee, Bonwoo (이본우, 수리과학과 2017학번)
3rd prize (Bronze): Ko, Sunghun (고성훈, 2018학번)

이종원 (수리과학과 2014학번) 40/40
채지석 (수리과학과 2016학번) 35/40
한준호 (수리과학과 2015학번) 35/40
이본우 (수리과학과 2017학번) 32/40
고성훈 (2018학번) 20/40
김태균 (수리과학과 2016학번) 19/40
최인혁 (물리학과 2015학번) 10/40
김건우 (수리과학과 2017학번) 8/40
최백규 (생명과학과 2016학번) 6/40
하석민 (수리과학과 2017학번) 6/40
길현준 (2018학번) 3/40
강한필 (전산학부 2016학번) 3/40
문정욱 (2018학번) 3/40
노우진 (물리학과 2015학번) 1/40
윤정인 (물리학과 2016학번) 1/40

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Solution: 2018-12 Property of Eigenvectors

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).

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Solution: 2018-11 Fallacy

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.

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Solution: 2018-10 Probability of making an acute triangle

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).

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Solution: 2018-09 Sum of digits

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).

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2018-11 Fallacy

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).

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Solution: 2018-08 Large LCM

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).

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Solution: 2018-07 A tridiagonal matrix

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).

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