ΗΜΕΡΑ 1η
1. For 
prove that
.
Prove that the line connect 2 center of 
and 
go through a fixed point. (where 
be a circumscribed circle of triangle 
).
4 Two people A and B play a game in the
grid (
). Each person respectively (A plays first) draw a segment between two point of the grid such that this segment doesn't contain any point (except the 2 ends) and also the segment (except the 2 ends) doesn't intersect with any other segments. The last person who can't draw is the loser. Which one (of A and B) have the winning tactics?
1. Let
be a positive integer. Suppose there exist non-negative integers 
such that 
. Prove that 
.
2. Let
defined by: 
, 
. Knowing that
(i)
.
(ii)
. Prove that 
.
and go through a fixed point. (where be a circumscribed circle of triangle ).4 Two people A and B play a game in the
grid (). Each person respectively (A plays first) draw a segment between two point of the grid such that this segment doesn't contain any point (except the 2 ends) and also the segment (except the 2 ends) doesn't intersect with any other segments. The last person who can't draw is the loser. Which one (of A and B) have the winning tactics? 1. Let
be a positive integer. Suppose there exist non-negative integers such that . Prove that . 2. Let
defined by: , . Knowing that(i)
.(ii)
. Prove that . 3. In a country, there are some cities and the city named Ben Song is capital. Each cities are connected with others by some two-way roads. One day, the King want to choose 
cities to add up with Ben Song city to establish an expanded capital such that the two following condition are satisfied:
(i) With every two cities in expanded capital, we can always find a road connecting them and this road just belongs to the cities of expanded capital.
(ii) There are exactly
cities which do not belong to expanded capital have the direct road to at least one city of expanded capital.
Prove that there are at most
options to expand the capital for the King.
cities to add up with Ben Song city to establish an expanded capital such that the two following condition are satisfied:(i) With every two cities in expanded capital, we can always find a road connecting them and this road just belongs to the cities of expanded capital.
(ii) There are exactly
cities which do not belong to expanded capital have the direct road to at least one city of expanded capital.Prove that there are at most
options to expand the capital for the King.
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