TR-MO 2003 Problem 3

Let $f: \mathbb R \rightarrow \mathbb R$ be a function such that

$f(tx_1+(1-t)x_2)\leq tf(x_1)+(1-t)f(x_2)$

for all $x_1 , x_2 \in \mathbb R$ and $t\in (0,1)$ . Show that

$\sum_{k=1}^{2003}f(a_{k+1})a_k \geq \sum_{k=1}^{2003}f(a_k)a_{k+1}$

for all real numbers $a_1,a_2,...,a_{2004}$ such that $a_1\geq a_2\geq ... \geq a_{2003}$ and $a_{2004}=a_1$