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Problem of the week

Consider a function \(f: \{1,2,\dots, n\}\rightarrow \mathbb{R}\) satisfying the following for all \(1\leq a,b,c \leq n-2\) with \(a+b+c\leq n\).

\[ f(a+b)+f(a+c)+f(b+c) - f(a)-f(b)-f(c)-f(a+b+c) \geq 0 \text{ and } f(1)=f(n)=0.\]

Prove or disprove this: all such functions \(f\) always have only nonnegative values on its domain.

Acknowledgement: This problem arises during a research discussion between June Huh, Jaehoon Kim and Matt Larson.

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