# 2017-16 Finding a rectangle

Is it possible to color all lattice points ($$\mathbb Z\times \mathbb Z$$) in the plane into two colors such that if four distinct points $$(a,b), (a+c,b), (a,b+d), (a+c,b+d)$$ have the same color, then $$d/c\notin \{1,2,3,4,6\}$$?

(The next POW problem will be posted on October 20. Happy Chuseok and good luck with your midterm exams.)

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# Solution: 2017-14 Polynomials of degree at most n

Let $$f(x)\in \mathbb R[x]$$ be a polynomial of degree at most $$n$$ such that $x^2+f(x)^2\le 1$ for all $$-1\le x\le 1$$. Prove that $$\lvert f'(x)\rvert \le 2(n-1)$$ for all $$-1\le x\le 1$$.

The best solution was submitted by Huy Tùng Nguyễn (수리과학과 2016학번). Congratulations!

Here is his solution of problem 2017-14.

Alternative solutions were submitted by 유찬진 (수리과학과 2015학번, +3), 이본우 (2017학번, +2). One incorrect solution was submitted.

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# 2017-15 Infinite product

For $$x \in (1, 2)$$, prove that there exists a unique sequence of positive integers $$\{ x_i \}$$ such that $$x_{i+1} \geq x_i^2$$ and
$x = \prod_{i=1}^{\infty} (1 + \frac{1}{x_i}).$

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# Solution: 2017-13 Infinite series with recurrence relation

Let $$a_0 = a_1 =1$$ and $$a_n = n a_{n-1} + (n-1) a_{n-2}$$ for $$n \geq 2$$. Find the value of
$\sum_{n=0}^{\infty} (-1)^n \frac{n!}{a_n a_{n+1}}.$

The best solution was submitted by Choi, Daebeom (최대범, 수리과학과 2016학번). Congratulations!

Here is his solution of problem 2017-13.

Alternative solutions were submitted by 국윤범 (수리과학과 2015학번, +3), 김동률 (수리과학과 2015학번, +3), 김태균 (수리과학과 2016학번, +3), 유찬진 (수리과학과 2015학번, +3), 윤준기 (전기및전자공학부 2014학번, +3), 이본우 (2017학번, +3), 이태영 (수리과학과 2013학번, +3), 장기정 (수리과학과 2014학번, +3, solution), 조태혁 (수리과학과 2014학번, +3, solution), 최인혁 (물리학과 2015학번, +3), Huy Tung Nguyen (수리과학과 2016학번, +3), 김기택 (수리과학과 2015학번, +2), 이재우 (함양고등학교 2학년, +2), 정의현 (수리과학과 2015학번, +2).

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# 2017-14 Polynomials of degree at most n

Let $$f(x)\in \mathbb R[x]$$ be a polynomial of degree at most $$n$$ such that $x^2+f(x)^2\le 1$ for all $$-1\le x\le 1$$. Prove that $$\lvert f'(x)\rvert \le 2(n-1)$$ for all $$-1\le x\le 1$$.

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# 2017-13 Infinite series with recurrence relation

Let $$a_0 = a_1 =1$$ and $$a_n = n a_{n-1} + (n-1) a_{n-2}$$ for $$n \geq 2$$. Find the value of
$\sum_{n=0}^{\infty} (-1)^n \frac{n!}{a_n a_{n+1}}.$

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# Solution: 2017-12 Invertible matrices

Let $$A$$ and $$B$$ be $$n\times n$$ matrices. Prove that if $$n$$ is odd and both $$A+A^T$$ and $$B+B^T$$ are invertible, then $$AB\neq 0$$.

The best solution was submitted by Shin, Joonhyung (신준형, 수리과학과 2015학번). Congratulations!

Here is his solution of the problem 2017-12.

Alternative solutions were submitted by 국윤범 (수리과학과 2015학번, +3), 김동률 (수리과학과 2015학번, +3), 김태균 (수리과학과 2016학번, +3), 위성군 (수리과학과 2015학번, +3), 유찬진 (수리과학과 2015학번, +3), 윤준기 (전기및전자공학부 2014학번, +3), 이본우 (2017학번, +3), 이준협 (하나고등학교, +3), 이태영 (수리과학과 2013학번, +3), 이형진 (청주대 수학교육과 2011학번, +3), 임성혁 (수리과학과 2016학번, +3), 장기정 (수리과학과 2014학번, +3), 조태혁 (수리과학과 2014학번, +3), 최대범 (수리과학과 2016학번, +3), 최인혁 (물리학과 2015학번, +3), Huy Tung Nguyen (수리과학과 2016학번, +3), Saba Dzmanashvili (2017학번, +3).

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Let $$A$$ and $$B$$ be $$n\times n$$ matrices. Prove that if $$n$$ is odd and both $$A+A^T$$ and $$B+B^T$$ are invertible, then $$AB\neq 0$$.