▪ Vietnam Team Selection Tests 2012

Ημέρα 1η

1. Consider a circle and two fixed points on such that is not the diameter of . is an arbitrary point on , distinct from . Let be the midpoints of , respectively, be the feet of perpendiculars from to , to , to , respectively. The two tangents at to the circumcircle of triangle meet at . Prove that is a fixed point (as moves on ). 

2. Consider a rectangular grid with rows and columns. There are water fountains on some of the squares. A water fountain can spray water onto any of it's adjacent squares, or a square in the same column such that there is exactly one square between them. Find the minimum number of fountains such that each square can be sprayed in the case that

a) ;

b) .

3. Let be a prime. Prove that is the largest positive integer which satisfies the following condition:

For any integers such that is not divisible by and is divisible by , there exists integers belonging to the set such that is divisible by . 

Ημέρα 2η

1. Consider the sequence where and for all . Prove that 

 

is a perfect square. 

2. Prove that is the largest constant such that if there exist positive numbers satisfying:

then for every such that , we have that are sides of a triangle. 

3. There are students taking part in the Team Selection Test. It is known that every student knows exactly other students. Show that we can divide the students into groups or .

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