a short but very puzzling problem

Consider n functions f_1..f_n, all are from [0, inf) to [0, inf),
continuous, strictly increasing, and f_i(0)=0 for all i. Prove or
disprove that for all c<1, it is impossible that
1. [f_1(x)+...+f_n(x)] / n<cx for all x>=0, and
2. [f_1^{-1}(x)+...+f_n^{-1}(x)] / n<cx for all x>=0.

Here -1 means inverse, not reciprocal.
I can only solve the case n<=2. I don't know the answer for the general
case, it came from my research.
Solutions, or pointers to references would be greatly appreciated.
Best,
Qiqi Yan from SJTU.

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