▪ Iran Team Selection Test 2012

1. Find all positive integers such that for all integers that , and

   

have same parity. 

2. Consider is circumcircle of an acute triangle . is midpoint of arc and is incenter of triangle . Let intersect in and for second time in . Let be a point on line such that is parallel to . Prove that is bisector of angle . 

3. Let be a positive integer. Let be a subset of points on the plane with these conditions: 

There does not exist lines in the plane such that every element of be on at least one of them. 

for all there exists lines in the plane such that every element of be on at least one of them.

Find maximum of .