Prove that there exist infinitely many pairs of positive integers \( (m, n) \) satisfying the following properties:

(1) gcd\( (m, n) = 1 \).

(2) \((x+m)^3 = nx\) has three distinct integer solutions.

The best solution was submitted by 이종원. Congratulations!

Alternative solutions were submitted by 김경석 (+3), 김은혜 (+3), 김일희 (+3), 김찬민 (+3), 박훈민 (+3), 안현수 (+3), 어수강 (+3), 윤성철 (+3), 이영민 (+3), 장경석 (+3), 장기정 (+3), 정성진 (+3), 조준영 (+3), 채석주 (+3), 황성호 (+3), 박경호 (+2), 조남경 (+2). An incorrect solutions was submitted by N.J.H. (Some initials here might have been improperly chosen.)