# Solution: 2015-22 An integral

Evaluate the following integral $\int_0^\infty \frac{\cos cx}{(\cosh x)^2} \, dx$ for a real constant $$c$$.

The best solution was submitted by Sunghyuk Park (박성혁, 수리과학과 2014학번). Congratulations!

Here is his solution of problem 2015-22.

Alternative solutions were submitted by 신준형 (2015학번, +3), 이종원 (수리과학과 2014학번, +3), 장기정 (수리과학과 2014학번, +3), 유찬진 (2015학번, +2), 최인혁 (2015학번, +2), 이예찬 (오송고등학교 교사, +2), Luis F. Abanto-Leon (+2).

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# 2015-23 Fixed points

Let $$f:[0,1)\to[0,1)$$  be a function such that $f(x)=\begin{cases} 2x,&\text{if }0\le 2x\lt 1,\\ 2x-1, & \text{if } 1\le 2x\lt 2.\end{cases}$ Find all $$x$$ such that $f(f(f(f(f(f(f(x)))))))=x.$

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# Solution: 2015-21 Differentiable function

Assume that a function $$f : (0, 1) \to [0, \infty)$$ satisfies $$f(x) = 0$$ at all but countably many points $$x_1, x_2, \cdots$$. Let $$y_n = f(x_n)$$. Prove that, if $$\sum_{n=1}^{\infty} y_n < \infty$$, then $$f$$ is differentiable at some point.

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

Here is his solution of problem 2015-21.

Alternative solutions were submitted by 박성혁 (수리과학과 2014학번, +3), 이종원 (수리과학과 2014학번, +3), 최인혁 (2015학번, +3), 신준형 (2015학번, +2).

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# 2015-22 An integral

Evaluate the following integral $\int_0^\infty \frac{\cos cx}{(\cosh x)^2} \, dx$ for a real constant $$c$$.

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# Solution: 2015-20 Dense function

Prove or disprove the following statement:
There exists a function $$f : \mathbb{R} \to \mathbb{R}$$ such that
(1) $$f \equiv 0$$ almost everywhere, and
(2) for any nonempty open interval $$I$$, $$f(I) = \mathbb{R}$$.

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

Here is his solution of problem 2015-20.

Alternative solutions were submitted by 박성혁 (수리과학과 2014학번, +3, his solution), 이영민 (수리과학과 2012학번, +3), 이종원 (수리과학과 2014학번, +3, his solution), 장기정 (수리과학과 2014학번, +3, his solution), 최인혁 (2015학번, +3), 김동률 (2015학번, +2), 이신영 (물리학과 2012학번, +2),  송교범 (서대전고등학교 2학년, +3).

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# 2015-21 Differentiable function

Assume that a function $$f : (0, 1) \to [0, \infty)$$ satisfies $$f(x) = 0$$ at all but countably many points $$x_1, x_2, \cdots$$. Let $$y_n = f(x_n)$$. Prove that, if $$\sum_{n=1}^{\infty} y_n < \infty$$, then $$f$$ is differentiable at some point.

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# Solution: 2015-19 Sum of tangent functions

Evaluate $\sum_{k=0}^{n} 2^k \tan \frac{\pi}{4\cdot 2^{n-k}}.$

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

Here is his solution of problem 2015-19.

Alternative solutions were submitted by 김기택 (2015학번, +3), 박성혁 (수리과학과 2014학번, +3, his solution), 박훈민 (수리과학과 2013학번, +3), 신준형 (2015학번, +3), 이상민 (수리과학과 2014학번, +3), 이영민 (수리과학과 2012학번, +3), 이종원 (수리과학과 2014학번, +3), 최인혁 (2015학번, +3), Luis F. Abanto-Leon (+3), 김강식 (포항공대 수학과 2013학번, +3), 엄태강 (포항공대 수학과 2014학번, +3), 임준휘 (포항공대 수학과 2014학번, +3).

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There exists a function $$f : \mathbb{R} \to \mathbb{R}$$ such that
(1) $$f \equiv 0$$ almost everywhere, and
(2) for any nonempty open interval $$I$$, $$f(I) = \mathbb{R}$$.