Mathematica competitions in Croatia

In some country there are $c$ cities and $r$ roads, every road connects two different cities and between any two cities there is at most one road. Roads are named by numbers $1,2,\dots ,r$. Tonči travels along some roads in such a way that, when he writes down the names of the roads in the order he passes through them, he obtains an ascending sequence of numbers. Show that there is a city such that starting from it Tonči can pass through at least $\frac{2r}{c}$ roads.