# Solution: 2015-15 A sequence periodic modulo m for all m

Does there exist an infinite sequence such that (i) every integer appears infinitely many times and (ii) the sequence is periodic modulo $$m$$ for every positive integer $$m$$?

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

Here is his solution of problem 2015-15.

Alternative solutions were submitted by 이종원 (수리과학과 2014학번, +3), 신준형 (2015학번, +3), 최인혁 (2015학번, +2), 이영민 (수리과학과 2012학번, +2), 장기정 (수리과학과 2014학번, +2).

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# 2015-16 Complex integral

Evaluate the following integral for $$z \in \mathbb{C}^+$$.

$\frac{1}{2\pi} \int_{-2}^2 \log (z-x) \sqrt{4-x^2} dx.$

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# Solution: 2015-14 Local and absolute maximum

Find all positive integers $$n$$ such that the following statement holds:

Let $$f:\mathbb{R}^n\to \mathbb {R}$$ be a differentiable function that has a unique critical point $$c$$. If $$f$$ has a local maximum at $$c$$, then $$f(c)$$ is an absolute maximum of $$f$$.

The best solution was submitted by Choi, Inhyeok (최인혁, 2015학번). Congratulations!

Here is his solution of problem 2015-14.

Alternative solutions were submitted by 이종원 (수리과학과 2014학번, +3), 김재준 (2014학번, +3), 박훈민 (수리과학과 2013학번, +3), 박성혁 (수리과학과 2014학번, +3), 오동우 (2015학번, +3), 이영민 (수리과학과 2012학번, +3), 장기정 (수리과학과 2014학번, +3), 신준형 (2015학번, +2). One incorrect solutions were received (LAL). Delayed submissions were not graded.

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# 2015-15 A sequence periodic modulo m for all m

Does there exist an infinite sequence such that (i) every integer appears infinitely many times and (ii) the sequence is periodic modulo $$m$$ for every positive integer $$m$$?

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# 2015-14 Local and absolute maximum

Find all positive integers $$n$$ such that the following statement holds:

Let $$f:\mathbb{R}^n\to \mathbb {R}$$ be a differentiable function that has a unique critical point $$c$$. If $$f$$ has a local maximum at $$c$$, then $$f(c)$$ is an absolute maximum of $$f$$.

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# Solution: 2015-13 Minimum

Find the minimum value of
$\int_{\mathbb{R}} f(x) \log f(x) dx$
among functions $$f: \mathbb{R} \rightarrow \mathbb{R}^+ \cup \{ 0 \}$$ that satisfy the condition
$\int_{\mathbb{R}} f(x) dx = \int_{\mathbb{R}} x^2 f(x) dx = 1.$

The best solution was submitted by Lee, Jongwon (이종원, 수리과학과 2014학번). Congratulations!

Here is his solution of problem 2015-13.

Alternative solutions were submitted by 김경석 (2015학번, +3), 김재준 (2014학번, +3), 김희주 (2015학번, +2), 박성혁 (수리과학과 2014학번, +2), 박훈민 (수리과학과 2013학번, +3), 신준형 (2015학번, +3), 오동우 (2015학번, +2), 이신영 (물리학과 2012학번, +2), 이영민 (수리과학과 2012학번, +2), 이정환 (2015학번, +3), 장기정 (수리과학과 2014학번, +2), 최인혁 (2015학번, +2), Luis F. Abanto-Leon (+2), 이시우 (포항공대 수학과 2013학번, +3). Two incorrect solutions (L.S.M., H.I.S.) were submitted.

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$\int_{\mathbb{R}} f(x) \log f(x) dx$
among functions $$f: \mathbb{R} \rightarrow \mathbb{R}^+ \cup \{ 0 \}$$ that satisfy the condition
$\int_{\mathbb{R}} f(x) dx = \int_{\mathbb{R}} x^2 f(x) dx = 1.$