MathDB

There are

n

cities in a country, where

n>1

. There are railroads connecting some of the cities so that you can travel between any two cities through a series of railroads (railroads run in both direction.) In addition, in this country, it is impossible to travel from a city, through a series of distinct cities, and return back to the original city. We define the degree of a city as the number of cities directly connected to it by a single segment of railroad. For a city

A

that is directly connected to

x

cities, with

y

of those cities having a smaller degree than city

A

, the significance of city

A

is defined as

\frac{y}{x}

.
Find the smallest positive real number

t

so that, for any

n>1

, the sum of the significance of all cities is less than

tn

, no matter how the railroads are paved.
Proposed by houkai

Problem Statement