4

Part of 2013 Middle European Mathematical Olympiad

Problems(2)

Diophantine equation

Source: Middle European Mathematical Olympiad 2013 I-4

a

b

be positive integers. Prove that there exist positive integers

x

y

\binom{x+y}{2} = ax + by .

number theory proposednumber theory

points in the plane

Source: Middle European Mathematical Olympiad 2013 T-4

Consider finitely many points in the plane with no three points on a line. All these points can be coloured red or green such that any triangle with vertices of the same colour contains at least one point of the other colour in its interior. What is the maximal possible number of points with this property?

combinatorics proposedcombinatorics