# Solution: 2019-11 Smallest prime

Find the smallest prime number $$p \geq 5$$ such that there exist no integer coefficient polynomials $$f$$ and $$g$$ satisfying
$p | ( 2^{f(n)} + 3^{g(n)})$
for all positive integers $$n$$.

The best solution was submitted by 김태균 (수리과학과 2016학번). Congratulations!

Here is his solution of problem 2019-11.

Other solutions were submitted by 고성훈 (2018학번, +3), 조재형 (수리과학과 2016학번, +3), 채지석 (수리과학과 2016학번, +3), 최백규 (생명과학과 2016학번, +3).

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# Solution: 2018-06 Product of diagonals

Let $$A_1,A_2,A_3,\ldots,A_n$$ be the vertices of a regular $$n$$-gon on the unit circle. Evaluate $$\prod_{i=2}^n A_1A_i$$. (Here, $$A_1A_i$$ denotes the length of the line segment.)

The best solution was submitted by Taegyun Kim (김태균, 수리과학과 2016학번). Congratulations!

Here is his solution of problem 2018-06.

Alternative solutions were submitted by 권홍 (중앙대 물리학과, +3), 고성훈 (2018학번, +3), 김건우 (수리과학과 2017학번, +3), 이본우 (수리과학과 2017학번, +3), 이종원 (수리과학과 2014학번, +3), 채지석 (수리과학과 2016학번, +3), 최백규 (생명과학과 2016학번, +3), 하석민 (수리과학과 2017학번, +3), 한준호 (수리과학과 2015학번, +3).

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# Solution: 2017-19 Identity

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

The best solution was submitted by Kim, Taegyun (김태균, 수리과학과 2016학번). Congratulations!

Here is his solution of problem 2017-19.

Alternative solutions were submitted by 국윤범 (수리과학과 2015학번, +3), 길현준 (인천과학고등학교 2학년, +3), 민찬홍 (중앙대학교사범대학부속고등학교 3학년, +3), 유찬진 (수리과학과 2015학번, +3), 이본우 (2017학번, +3), 채지석 (2016학번, +3), 최대범 (수리과학과 2016학번, +3), Huy Tung Nguyen (수리과학과 2016학번, +3), 이재우 (함양고등학교 2학년, +2), 하석민 (2017학번, +2), Saba Dzmanashvili & Mirali Ahmadili  (2017학번, +2).

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# Concluding 2016 Fall

Thanks all for participating POW actively. Here’s the list of winners:

1st prize (Gold): Shin, Joonhyung (신준형, 수리과학과 2015학번)
2nd prize (Silver): Jang, Kijoung (장기정, 수리과학과 2014학번).
2nd prize (Silver): Kim, Taegyun (김태균, 수리과학과 2014학번).
2nd prize (Silver): Kook, Yun Bum (국윤범, 수리과학과 2015학번).
3rd prize (Bronze): Lee, Sangmin (이상민, 수리과학과 2014학번).
3rd prize (Bronze): Lee, Jongwon (이종원, 수리과학과 2014학번).

신준형 (수리과학과 2015학번) 32, 장기정 (수리과학과 2014학번) 31, 김태균 (2016학번) 30, 국윤범 (수리과학과 2015학번) 29, 이상민 (수리과학과 2014학번) 19, 이종원 (수리과학과 2014학번) 19, 최대범 (2016학번) 16, 윤준기 (전기및전자공학부 2014학번) 14, 최인혁 (물리학과 2015학번) 13, 채지석 (2016학번) 12, 김재현 (2016학번) 11, 이정환 (수리과학과 2015학번) 9, Ivan Adrian Koswara (전산학부 2013학번) 6, 강한필 (2016학번) 6, 위성군 (수리과학과 2015학번) 6, 김기택 (수리과학과 2015학번) 6, 박기연 (2016학번) 5, 한준호 (수리과학과 2015학번) 5, 조준영 (수리과학과 2012학번) 3, 박현준 (물리학과 2014학번) 3, 오동우 (2015학번) 3, 유찬진 (수리과학과 2015학번) 3, 임성혁 (2016학번) 3, Muhammaadfiruz Hasanov (2014학번) 3, 정의현 (수리과학과 2015학번) 2, 박진호 (물리학과 2015학번) 2, 정성진 (수리과학과 2013학번) 2.

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# Solution: 2016-23 Inequality on complex numbers

Suppose that $$z_1, z_2, \dots, z_n$$ are complex numbers satisfying $$\sum_{k=1}^n z_k = 0$$. Prove that
$\sum_{k=1}^n |z_{k+1} – z_k|^2 \geq 4 \sin^2 \left( \frac{\pi}{n} \right) \sum_{k=1}^n |z_k|^2,$
where we let $$z_{n+1} = z_1$$.

The best solution was submitted by Kim, Taegyun (김태균, 2016학번). Congratulations!

Here is his solution of problem 2016-23.

Alternative solutions were submitted by 신준형 (수리과학과 2015학번, +3), 장기정 (수리과학과 2014학번, +3, alternative solution), 국윤범 (수리과학과 2015학번, +3), 김기현 (수리과학과 대학원생, +3, alternative solution), 이상민 (수리과학과 2014학번, +2). One incorrect solution was submitted.

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For a positive integer $$n$$, let $$d(n)$$ be the number of positive divisors of $$n$$. Prove that, for any positive integer $$M$$, there exists a constant $$C>0$$ such that $$d(n) \geq C ( \log n )^M$$ for infinitely many $$n$$.