▪ Armenia Team Selection Test 2011

Ημέρα 1η

Let be positive integer and let be numbers which make a geometrical progression. Prove that the system of the equations

have solutions in real numbers if and only if 

i) and is not equal 0; where , , or

ii) and if , or

iii) , where .

Solve the system of equations for , 

2 Find all functions satisfying for all real values . 

3 We have square. In this square there are dominoes of the sides which are in square randomly. Prove that we always can add 1 domino. 

4 Prove that for all positive real numbers

Prove that equality doesn't hold. 

Ημέρα 2η

1 Let be a convex pentagon such that and degrees. Prove that .

2 Inside of a triangle there is a convex polygon . Prove that

3 Inside of a equilateral triangle there is a any point. are feet of the perpendiculars from to respectively. are the midpoints of respectively. Prove that meet at one point if and only if when meet at one point. 

4 Let are the midpoints of respectively. Consider that are points on such that .Let are the incenters of the triangles respectively and let be the centroids of the triangles respectively. Prove that . 

Ημέρα 3η

1 Find all pairs of integers for which . 

2 Prove that there exists integers such that . 

4 Prove that for any positive integer ,

i) doesn't have any prime divisor less that .

ii) has a prime divisor such that .