MathDB

Two sequences

b_i

c_i

0 \le i \le 100

contain positive integers, except

c_0=0

and

b_{100}=0

. Some towns in Graphland are connected with roads, and each road connects exactly two towns and is precisely

1

km long. Towns, which are connected by a road or a sequence of roads, are called neighbours. The length of the shortest path between two towns

X

and

Y

is denoted as distance. It is known that the greatest distance between two towns in Graphland is

100

km. Also the following property holds for every pair

X

and

Y

of towns (not necessarily distinct): if the distance between

X

and

Y

is exactly

k

km, then

Y

has exactly

b_k

neighbours that are at the distance

k+1

from

X

, and exactly

c_k

neighbours that are at the distance

k-1

from

X

. Prove that

\frac{b_0b_1 \cdot \cdot \cdot b_{99}}{c_1c_2 \cdot \cdot \cdot c_{100}}

is a positive integer.

Problem Statement