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).