A country is called a "pentagonal" if every city in it is connected by air routes to exactly five other cities (there are no international flights).

a) Draw a diagram of air routes for a pentagonal country with 10 cities.

b) How many air routes are there in a pentagonal country with 50 cities?

c) Can there be a pentagonal country with exactly 46 air routes?
a) In a regular decagon, we draw all sides, all small diagonals (connecting vertices one apart), and all large diagonals (connecting opposite vertices).

b) $50 \cdot 5: 2=125$

c) If a country has $n$ cities, then $46=5 n / 2$. But 46 is not divisible by 5.

## Answer

b) 125 air routes; c) it cannot.