Math Olympiad-2

### Author Topic: Math Olympiad-2  (Read 1687 times)

#### Mosammat Arifa Akter

• Full Member
• Posts: 187
• Test
##### Math Olympiad-2
« on: April 21, 2015, 11:39:54 PM »
2. There are n cities in the country. Between any two cities there is at most one road. Suppose that the total number of roads is n. Prove that there is a city such that starting from there it is possible to come back to it without ever traveling the same road twice.

Answer: Let therebe C ways to take n points such that it is not possible to return to its original position. So, we can take 1 point in (n−1). 2 points be taken in (n−1)(n−2) ........... So n points can be taken (n−1)(n−2)(n−3)(n−n+1)(n−n)
C=(n−1)(n−2)(n−n+1)(n−n)=0.
So, there is 0 ways to take n points such that it is not possible to return to its original position. So,it is always possible to return to its original position.
Mosammat Arifa Akter
Senior Lecturer(Mathematics)
General Educational Development
Daffodil International University