▪ 29η Βαλκανική Μαθηματική Ολυμπιάδα 2012

Τα θέματα της 29ης Βαλκανιάδας Μαθηματικών, που έγινε σήμερα Σάββατο 28 Απριλίου, στην Αττάλεια της Τουρκίας.

Πρόβλημα 1ο

Let , and be points lying on a circle with centre . Assume that . Let be the point of intersection of the line with the line perpendicular to at . Let be the line through which is perpendicular to . Let be the point of intersection of with the line , and let be the point of intersection of with that lies between and .

Prove that the circumcircles of triangles and are tangent at .
Κάντε κλικ στον σύνδεσμο να δείτε τη λύση της άσκησης: Φραγκάκης Νίκος(Doloros)

Πρόβλημα 2ο 

Prove that 

for all positive real numbers and . 

Πρόβλημα 3ο 

Let be a positive integer. Let For each subset of , we write for the sum of all elements of , with the convention that where is the empty set. Suppose that is a real number with

Prove that there is a subset of such that

  .

Πρόβλημα 4ο 

Let be the set of positive integers. Find all functions such that the following conditions both hold:

for every positive integer ,

divides whenever and are different positive integers.

Δείτε εδώ τα θέματα στα Ελληνικά (pdf).