TR-MO 1995 Problem 1

Let $m_{1},m_{2},\ldots,m_{k}$ be integers with $2\leq m_{1}$ and $2m_{1}\leq m_{i+1}$ for all $i$ . Show that for any integers $a_{1},a_{2},\ldots,a_{k}$ there are infinitely many integers $x$ which do not satisfy any of the congruences $$x\equiv a_{i}\ (\bmod \ m_{i}),\ i=1,2,\ldots k.$$