Prove or disprove the following:

There exist an infinite sequence of functions \( f_n: [0, 1] \to \mathbb{R} , n=1, 2, \dots \) ) such that

(1) \( f_n(0) = f_n(1) = 0 \) for any \( n \),

(2) \( f_n(\frac{a+b}{2}) \leq f_n(a) + f_n(b) \) for any \( a, b \in [0, 1] \),

(3) \( f_n – c f_m \) is not identically zero for any \( c \in \mathbb{R} \) and \( n \neq m \).

The best solution was submitted by 김기택 (수리과학과 대학원생, +4). Congratulations!

Here is the best solution of problem 2021-13.

Other solutions were submitted by 강한필 (전산학부 2016학번, +3), 고성훈 (수리과학과 2018학번, +3), 김민서 (수리과학과 2019학번, +3), 박정우 (수리과학과 2019학번, +3), 신주홍 (수리과학과 2020학번, +3), 이도현 (수리과학과 2018학번, +3), 이본우 (수리과학과 2017학번, +3), 이호빈 (수리과학과 대학원생, +3).