Problems

(You can suggest your solutions in the comment section)

Problem # 1

Han Xin counts his soldiers. When the soldiers arranged 3 by 3, there were 2 remaining soldiers. When they lined up 5 by 5, there were 4 remaining soldiers. When they lined up 7 by 7, there were 6 remaining soldiers. How much soldiers does Han Xin have ( minimum ) ?


Problem # 2


The Garden of a palace contains seven bridges, the king of the palace asks his children whether it is possible to cross each bridge in one and only once. As none of the king's children know the answer, the king has found you to tell him the answer. (Here is the plan of the garden )

Problem # 3

Prove that this fraction  21n+414n+3 is theis the most simplified whatever n is. ( n∈ℕ )$\mathbb{}$. 
First International Mathematical Olympiad's, first question ! Hint : use gcd
Problem # 4

If n friends were shaking hands. Will there be 2 people that shakes the same amount of hands ? ( to prove )

Problem # 5

Here we have a adjacent list ( a list that tells you the two connected intersections ) :

1 connected to 2
2  connected to 3
2  connected to  4
4  connected to 5
4  connected to 6
5  connected to 7
6  connected to 7
7  connected to 8

How many paths are there to the last intersection number 8 ?

Problem # 6

If a school has three people that has exactly the same anniversary. How many people are there in this school ( we want the minimum ) ?