Suppose that \(f\) is differentiable and \[ \lim_{x\to\infty} (f(x)+f'(x))=2.\]  What is \( \lim_{x\to\infty} f(x)\)?

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

Here is his solution of problem 2017-18.

Alternative solutions were submitted by 국윤범 (수리과학과 2015학번, +3), 김태균 (수리과학과 2016학번, +3), 민찬홍 (중앙대학교사범대학부속고등학교 3학년, +3), 이본우 (2017학번, +3), 장기정 (수리과학과 2014학번, +3, alternative solution), 채지석 (2016학번, +3), 최대범 (수리과학과 2016학번, +3), 하석민 (2017학번, +3), Huy Tung Nguyen (수리과학과 2016학번, +3), 윤준기 (전기및전자공학부 2014학번, +2). One incorrect solution was received.

For an integer \( n \geq 3 \), evaluate
\[
\inf \left\{ \sum_{i=1}^n \frac{x_i^2}{(1-x_i)^2} \right\},
\]
where the infimum is taken over all \( n \)-tuple of real numbers \( x_1, x_2, \dots, x_n \neq 1 \) satisfying that \( x_1 x_2 \dots x_n = 1 \).

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

Here is his solution of problem 2017-17.

Alternative solutions were submitted by 국윤범 (수리과학과 2015학번, +3), 김태균 (수리과학과 2016학번, +3), 장기정 (수리과학과 2014학번, +3), Huy Tung Nguyen (수리과학과 2016학번, +3), 김기택 (수리과학과 2015학번, +2), 유찬진 (수리과학과 2015학번, +2), 윤준기 (전기및전자공학부 2014학번, +2), 이본우 (2017학번, +2). One incorrect solution was received.

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
\[
x = \prod_{i=1}^{\infty} (1 + \frac{1}{x_i}).
\]

The best solution was submitted by Jang, Kijoung (장기정, 수리과학과 2014학번). Congratulations!

Here is his solution of problem 2017-15.

Alternative solutions were submitted by 국윤범 (수리과학과 2015학번, +3), 김기택 (수리과학과 2015학번, +3), 김태균 (수리과학과 2016학번, +3), 송교범 (고려대 수학과 2017학번, +3), 어수강 (서울대학교 수학교육과 박사과정, +3), 유찬진 (수리과학과 2015학번, +3), 윤준기 (전기및전자공학부 2014학번, +3), 이본우 (2017학번, +3), 이태영 (수리과학과 2013학번, +3), 조태혁 (수리과학과 2014학번, +3), 최대범 (수리과학과 2016학번, +3), Huy Tung Nguyen (수리과학과 2016학번, +3), 김동률 (수리과학과 2015학번, +2), 이재우 (함양고등학교 2학년, +2).

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

Here is his solution of problem 2017-16.

Alternative solutions were submitted by 국윤범 (수리과학과 2015학번, +3), 김태균 (수리과학과 2016학번, +3), 유찬진 (수리과학과 2015학번, +3), 이수환 (수리과학과 2011학번, +3), 이재우 (함양고등학교 2학년, +3), 장기정 (수리과학과 2014학번, +3), Dung Nguyen (전산학부 2015학번, +3), Huy Tung Nguyen (수리과학과 2016학번, +3).

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).

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).

Find the value of
\[
\sum_{n=1}^{\infty} \frac{1+ \frac{1}{2} + \dots + \frac{1}{n}}{n(2n-1)}.
\]

The best solution was submitted by Jo, Tae Hyouk (조태혁, 수리과학과 2014학번). Congratulations!

Here is his solution of problem 2017-11.

Alternative solutions were submitted by Huy Tung Nguyen (2016학번, +3), 최대범 (수리과학과 2016학번, +3), 이본우 (2017학번, +2).

This was the last problem of Spring 2017. Thank you for participating POW actively.

Let \(A\), \(B\) be matrices over the reals with \(n\) rows. Let \(M=\begin{pmatrix}A  &B\end{pmatrix}\). Prove that \[ \det(M^TM)\le \det(A^TA)\det(B^TB).\]

The best solution was submitted by Lee, Bonwoo (이본우, 17학번). Congratulations!

Here is his solution of problem 2017-10.

Alternative solutions were submitted by Huy Tung Nguyen (2016학번, +3), 조태혁 (수리과학과 2014학번, +3), 장기정 (수리과학과 2014학번, +3), 최대범 (수리과학과 2016학번, +2). One incorrect solution was received.