Question 70
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.
$$
\frac{b_0 b_1 \dots b_{99}}{c_1 c_2 \dots c_{100}}
$$
is an integer.