▪ Moldova Junior Balkan Team Selection Tests 2012

Ημέρα 1η

1. Let and are different integers, find the minimum value of the expression

   

and prove that it is minimum.
2. Let be positive real numbers, prove the inequality:

.

3. Let be an equilateral triangle, take line such that and passes through .
Let point be on side , the bisector of angle intersects line in point .
Prove that 

. 

4. Let there be an infinite sequence with defined by:

and , .

a) Does belong to the sequence?
b) Prove that the sequence doesn't contain perfect squares.

Ημέρα 2η

1. Find a sequence of distinct integers bigger than such that their sum is a perfect square and their product is a perfect cube.
2. Let be positive real numbers and . Prove that there exists a positive integer such that

.

3. Let be an isosceles triangle with . Take points on side and on side and the intersection of bisectors of angles and such that lies on side . Prove that is the midpoint of .
4. How many solutions does the system have:

where are non-zero integers.

H λύση της 3ης άσκησης της 1ης ημέρας: Φραγκάκης Νίκος (Doloros)