# 2017-05 Inequality for a continuous function

Suppose that $$f : (2, \infty) \to (-2, 2)$$ is a continuous function and there exists a positive constant $$m$$ such that $$| 1 + xf(x) + (f(x))^2 | \leq m$$ for any $$x > 2$$. Prove that, for any $$x > 2$$,
$\left| f(x) – \frac{\sqrt{x^2 -4}-x}{2} \right| \leq 6 \sqrt{m}.$

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# 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 수학과 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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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
$\gamma_1 + \gamma_2 + \dots + \gamma_k \leq (\alpha_1 + \alpha_2 + \dots + \alpha_k) + (\beta_1 + \beta_2 + \dots + \beta_k)$