▪ Poland Mathematical Olympiad Second Round 2012

1η Ημέρα

1. , solve the system of equations:

2. Prove that for tetrahedron ; vertex , center of insphere and centroid of are collinear iff areas of triangles are equal. 

3. Let be such numbers that set contains exactly different prime numbers. Prove that if we choose any different numbers from then we can find number from choosen numbers, which divide product of other numbers. 

2η Ημέρα

1. find all satisfying :

2. Let be a triangle with and , -incenter, -circumcenter. Prove that perpendicular bisector of , line and line  have a common point. 

3. Denote by the sum of the digits in the decimal representation of . Prove that there are infinitely many such that:

  .