2017-21 Maclaurin series

Prove or disprove the following statement: There exists a function whose Maclaurin series converges at only one point.

2017-20 Convergence of a series

Determine whether or not the following infinite series converges. \[ \sum_{n=0}^{\infty} \frac{ 1 }{2^{2n}} \binom{2n}{n}.\]

Solution: 2017-18 Limit

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.

Solution: 2017-17 An infimum

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.

2017-19 Identity

For an integer \( p \), define
\[
f_p(n) = \sum_{k=1}^n k^p.
\]
Prove that
\[
\frac{1}{2} \sum_{n=1}^{\infty} \frac{f_{-1}(n)}{f_3(n)} + 2\sum_{n=1}^{\infty} \frac{f_{-1}(n)}{f_1(n)} = \sum_{n=1}^{\infty} \frac{(f_{-1}(n))^2}{f_1(n)}.
\]

2017-18 Limit

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

2017-17 An infimum

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 \( x_1 x_2 \dots x_n = 1 \).

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

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).

Solution: 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 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).

2017-16 Finding a rectangle