Let $$f: [0, \infty) \to \mathbb{R}$$ be a function satisfying the following conditions:

(1) For any $$x, y \geq 0$$, $$f(x+y) \geq f(x)+f(y)$$.

(2) For any $$x \in [0, 2]$$, $$f(x) \geq x^2 – x$$.

Prove that, for any positive integer $$M$$ and positive reals $$n_1, n_2, \cdots, n_M$$ with $$n_1 + n_2 + \cdots + n_M = M$$, we have

$f(n_1) + f(n_2) + \cdots + f(n_M) \geq 0.$

The best solution was submitted by 박훈민. Congratulations!

Alternative solutions were submitted by 김경석 (+3), 김일희 (+3), 안현수 (+3), 어수강 (+3), 윤성철 (+3), 이영민 (+3), 이종원 (+3), 장경석 (+3), 장기정 (+3), 정성진 (+3), 정진야 (+3), 조준영 (+3), 채석주 (+3), 황성호 (+3). Incorrect solutions were submitted by K.W.J., K.H.S., N.J.H, M.K.Y., S.W.C., L.H.B., C.W.H. (Some initials here might have been improperly chosen.)

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