▪ Junior Balkan Mathematical Olympiad 2012

1. Let be positive real numbers such that . Prove that 

When does equality hold? 

2. Let the circles and intersect at two points and , and let be a common tangent of and that touches and at and respectively. If and , evaluate the angle .

3. On a board there are nails, each two connected by a rope. Each rope is colored in one of given distinct colors. For each three distinct colors, there exist three nails connected with ropes of these three colors.

a) Can be ?

b) Can be ? 

4 Find all positive integers and such that

  .

Δείτε εδώ τις επίσημες λύσεις των προβλημάτων.