# 2021-13 Not convex

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 ).

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