▪ Albania National Mathematical Olympiad 2012

1. Find all primes such that and are also primes. 

2. The trinomial is such that

   

has three real roots. Find the y-coordinate of the vertex of . 

3. Let be the sum of the first terms of the arithmetic sequence . Show that the value of the expression

does not depend on the numbers nor on the choice of the arithmetic sequence . 

4. Find all functions such that 

for all . 

5 Let be a triangle where . Let be the foot of the altitude taken from to ; and let be the orthocentre, the circumcentre of , and the point of intersection between the radius and the side . The midpoint of is .
a) Prove that the reflection of in is on the circumcircle of the triangle .
b) In what ratio does the segment divide the segment ?

Κάντε κλικ εδώ, για να κατεβάσετε τα θέματα και εδώ, τη λύση του 5ου θέματος, που μου έστειλε ο Φραγκάκης Νίκος (Doloros, από την Ιεράπετρα (2ο Λύκειο).