Baltic Way shortlist

Let $b_i$, $c_i$, $0\le i\le 100$ be two sequences of positive integers with two exceptions: $c_0=0$, $b_{100}=0$. Several villages are connected by roads, each road connects two villages which are called neighbours and has length $1$ km. Roads do not intersect each other, but can pass over/under each other. The distance between two villages $X$ and $Y$ is the length of the shortest path between them. In this country the maximal distance between two villages equals $100$ km and for every pair of villages $(X,Y)$ (the case $X=Y$ is allowed) the following condition holds: if distance between $X$ and $Y$ is $k$ km, then there are exactly $b_k$ ($c_k$, respectively) neighbours of $Y$ that are $1$ km further from (closer to, respectively) $X$ than $Y$. Show that the number $$\frac{b_0{b}_1\dots b_{99}}{c_1{c}_2\dots c_{100}}$$ is an integer.