RO-TST 2008 Problem 2

Let $a_i, b_i$ be positive real numbers, $i=1,2,\ldots,n$ , $n\geq 2$ , such that $a_i<b_i$ , for all $i$ , and also $$b_1+b_2+\cdots + b_n < 1 + a_1+\cdots + a_n.$$ Prove that there exists a $c\in\mathbb R$ such that for all $i=1,2,\ldots,n$ , and $k\in\mathbb Z$ we have $$(a_i+c+k)(b_i+c+k) > 0.$$