# Solution: 2017-03 Trigonometric equation

For an integer $$n \geq 4$$, find the solutions of the equation
$\sum_{k=1}^n \frac{\sin \frac{k\pi}{n+1}}{\sin (\frac{k\pi}{n+1} -x)} = 0.$

The best solution was submitted by Choi, Inhyeok (최인혁, 물리학과 2015학번). Congratulations!

Here is his solution of problem 2017-03.

Alternative solutions were submitted by 위성군 (수리과학과 2015학번, +3), 이본우 (2017학번, +3), 이시우 (포항공대 수학과 2013학번, +3), 장기정 (수리과학과 2014학번, +3), 조태혁 (수리과학과 2014학번, +3), 최대범 (수리과학과 2016학번, +3), Huy Tung Nguyen (2016학번, +3), 조정휘 (건국대학교 수학과 2014학번, +3), 배형진 (마포고 3학년, +2), 오동우 (수리과학과 2015학번, +2).

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# 2017-04 More than a half

Prove (or disprove) that exactly one of the following is true for every subset $$A$$ of $$\{ (i,j): i,j\in\{1,2,\ldots,n\}, i\neq j\}$$.

(i) There exists a sequence of distinct integers $$i_1,i_2,\ldots,i_k\in \{1,2,\ldots,n\}$$ for some integer $$k>1$$ such that $$(i_1,i_2), (i_2,i_3),\ldots,(i_{k-1},i_k), (i_k,i_1)\in A$$.

(ii) There exists a collection of finite sets $$A_1,A_2,\ldots,A_n$$ such that for all distinct $$i,j\in\{1,2,\ldots,n\}$$, $$(i,j)\in A$$ if and only if $$\lvert A_i\cap A_j\rvert > \frac12 \lvert A_i\rvert$$ and $$\lvert A_i\cap A_j\rvert \le \frac12 \lvert A_j\rvert$$

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# Solution: 2017-02 Low-degree polynomial

Let $$a_1,a_2,\ldots,a_n$$ be distinct points in $$\mathbb R^4$$. Does there exist a non-zero polynomial $$P(x_1,x_2,x_3,x_4)$$ such that
(1) the degree of $$P$$ is at most $$\lceil\sqrt{5} n^{1/4}\rceil$$ and
(2) $$P(a_i)=0$$ for all $$i=1,2,\ldots,n$$?

The best solution was submitted by You, Chanjin (유찬진, 수리과학과 2015학번). Congratulations!

Here is his solution of problem 2017-02.

Alternative solutions were submitted by 김태균 (수리과학과 2016학번, +3), 박지민 (전산학부 박사 2017학번, +3), 배형진 (마포고 3학년, +3), 송교범 (고려대 수학교육과 2017학번, +3), 오동우 (수리과학과 2015학번, +3), 위성군 (수리과학과 2015학번, +3), 이본우 (2017학번, +3), 이시우 (포항공대 수학과 2013학번, +3), 이준호 (2016학번, +3), 장기정 (수리과학과 2014학번, +3), 조태혁 (수리과학과 2014학번, +3), 최대범 (수리과학과 2016학번, +3), 최인혁 (물리학과 2015학번, +3), 홍혁표 (수리과학과 2013학번, +3), Huy Tung Nguyen (2016학번, +3).

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# 2017-03 Trigonometric equation

For an integer $$n \geq 4$$, find the solutions of the equation
$\sum_{k=1}^n \frac{\sin \frac{k\pi}{n+1}}{\sin (\frac{k\pi}{n+1} -x)} = 0.$

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# Solution: 2017-01 Eigenvalues of Hermitian matrices

Let $$A, B, C$$ be $$N \times N$$ Hermitian matrices with $$C = A+B$$. Let $$\alpha_1 \geq \dots \geq \alpha_N$$, $$\beta_1 \geq \dots \geq \beta_N$$, $$\gamma_1 \geq \dots \geq \gamma_N$$ be the eigenvalues of $$A, B, C$$, respectively. For any $$1 \leq k \leq N$$, prove that
$\gamma_1 + \gamma_2 + \dots + \gamma_k \leq (\alpha_1 + \alpha_2 + \dots + \alpha_k) + (\beta_1 + \beta_2 + \dots + \beta_k)$

The best solution was submitted by Sounggun Wee (위성군, 수리과학과 2015학번). Congratulations!

Here is his solution of problem 2017-01.

Alternative solutions were submitted by 강한필 (2016학번, +3), 김태균 (수리과학과 2016학번, +3), 배형진 (마포고 3학년, +3), 오동우 (수리과학과 2015학번, +3), 이시우 (포항공대 수학과 2013학번, +3), 이정환 (수리과학과 2015학번, +3), 장기정 (수리과학과 2014학번, +3), 조태혁 (수리과학과 2014학번, +3), 최대범 (수리과학과 2016학번, +3), 최인혁 (물리학과 2015학번, +3), Huy Tung Nguyen (2016학번, +3), 곽상훈 (수리과학과 2013학번, +3), 이본우 (2017학번, +3), 이태영 (수리과학과 2013학번, +2).

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# 2017-02 Low-degree polynomial

Let $$a_1,a_2,\ldots,a_n$$ be distinct points in $$\mathbb R^4$$. Does there exist a non-zero polynomial $$P(x_1,x_2,x_3,x_4)$$ such that
(1) the degree of $$P$$ is at most $$\lceil\sqrt{5} n^{1/4}\rceil$$ and
(2) $$P(a_i)=0$$ for all $$i=1,2,\ldots,n$$?

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# 2017-01 Eigenvalues of Hermitian matrices

Let $$A, B, C$$ be $$N \times N$$ Hermitian matrices with $$C = A+B$$. Let $$\alpha_1 \geq \dots \geq \alpha_N$$, $$\beta_1 \geq \dots \geq \beta_N$$, $$\gamma_1 \geq \dots \geq \gamma_N$$ be the eigenvalues of $$A, B, C$$, respectively. For any $$1 \leq k \leq N$$, prove that
$\gamma_1 + \gamma_2 + \dots + \gamma_k \leq (\alpha_1 + \alpha_2 + \dots + \alpha_k) + (\beta_1 + \beta_2 + \dots + \beta_k)$

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# Concluding 2016 Fall

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

1st prize (Gold): Shin, Joonhyung (신준형, 수리과학과 2015학번)
2nd prize (Silver): Jang, Kijoung (장기정, 수리과학과 2014학번).
2nd prize (Silver): Kim, Taegyun (김태균, 수리과학과 2014학번).
2nd prize (Silver): Kook, Yun Bum (국윤범, 수리과학과 2015학번).
3rd prize (Bronze): Lee, Sangmin (이상민, 수리과학과 2014학번).
3rd prize (Bronze): Lee, Jongwon (이종원, 수리과학과 2014학번).

신준형 (수리과학과 2015학번) 32, 장기정 (수리과학과 2014학번) 31, 김태균 (2016학번) 30, 국윤범 (수리과학과 2015학번) 29, 이상민 (수리과학과 2014학번) 19, 이종원 (수리과학과 2014학번) 19, 최대범 (2016학번) 16, 윤준기 (전기및전자공학부 2014학번) 14, 최인혁 (물리학과 2015학번) 13, 채지석 (2016학번) 12, 김재현 (2016학번) 11, 이정환 (수리과학과 2015학번) 9, Ivan Adrian Koswara (전산학부 2013학번) 6, 강한필 (2016학번) 6, 위성군 (수리과학과 2015학번) 6, 김기택 (수리과학과 2015학번) 6, 박기연 (2016학번) 5, 한준호 (수리과학과 2015학번) 5, 조준영 (수리과학과 2012학번) 3, 박현준 (물리학과 2014학번) 3, 오동우 (2015학번) 3, 유찬진 (수리과학과 2015학번) 3, 임성혁 (2016학번) 3, Muhammaadfiruz Hasanov (2014학번) 3, 정의현 (수리과학과 2015학번) 2, 박진호 (물리학과 2015학번) 2, 정성진 (수리과학과 2013학번) 2.

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# Solution: 2016-23 Inequality on complex numbers

Suppose that $$z_1, z_2, \dots, z_n$$ are complex numbers satisfying $$\sum_{k=1}^n z_k = 0$$. Prove that
$\sum_{k=1}^n |z_{k+1} – z_k|^2 \geq 4 \sin^2 \left( \frac{\pi}{n} \right) \sum_{k=1}^n |z_k|^2,$
where we let $$z_{n+1} = z_1$$.

The best solution was submitted by Kim, Taegyun (김태균, 2016학번). Congratulations!

Here is his solution of problem 2016-23.

Alternative solutions were submitted by 신준형 (수리과학과 2015학번, +3), 장기정 (수리과학과 2014학번, +3, alternative solution), 국윤범 (수리과학과 2015학번, +3), 김기현 (수리과학과 대학원생, +3, alternative solution), 이상민 (수리과학과 2014학번, +2). One incorrect solution was submitted.

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Let $$M_n=(a_{ij})_{ij}$$ be an $$n\times n$$ matrix such that $a_{ij}=\binom{2(i+j-1)}{i+j-1}.$ What is $$\det M_n$$?