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for all real a_1 , a_2 , ... , a_2004

Source: Turkey NMO 2003 Problem 3

February 24, 2009
functioninequalities unsolvedinequalities

Problem Statement

Let f:R→R f: \mathbb R \rightarrow \mathbb R be a function such that f(tx_1\plus{}(1\minus{}t)x_2)\leq tf(x_1)\plus{}(1\minus{}t)f(x_2) for all x1,x2∈R x_1 , x_2 \in \mathbb R and t∈(0,1) t\in (0,1). Show that \sum_{k\equal{}1}^{2003}f(a_{k\plus{}1})a_k \geq \sum_{k\equal{}1}^{2003}f(a_k)a_{k\plus{}1} for all real numbers a1,a2,...,a2004 a_1,a_2,...,a_{2004} such that a1≥a2≥...≥a2003 a_1\geq a_2\geq ... \geq a_{2003} and a_{2004}\equal{}a_1
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