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f(m x+(1- m)y) < m f{x)+(1- m)f(y), parallelogram on graph

Source: 1999 Romania NMO IX p4

August 15, 2024
geometryparallelogramalgebrainequalities

Problem Statement

a) Let a,b∈Ra,b\in R, a<ba <b. Prove that x∈(a,b)x \in (a,b) if and only if there exists λ∈(0,1)\lambda \in (0,1) such that x=λa+(1−λ)bx=\lambda a +(1-\lambda)b.
b) If the function f:R→Rf: R \to R has the property: f(λx+(1−λ)y)<λf(x)+(1−λ)f(y),∀x,y∈R,x≠y,∀λ∈(0,1),f (\lambda x+(1-\lambda) y) < \lambda f(x) + (1-\lambda)f(y), \forall x,y \in R, x\ne y, \forall \lambda \in (0,1), prove that one cannot find four points on the function’s graph that are the vertices of a parallelogram
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