▪ Centro American Mathematical Olympiad 2011

ΗΜΕΡΑ 1η

1. Consider a cube with a fly standing at each of its vertices. When a whistle blows, each fly moves to a vertex in the same face as the previous one but diagonally opposite to it. After the whistle blows, in how many ways can the flies change position so that there is no vertex with 2 or more flies?
2. In a scalene triangle , is the foot of the altitude through , is the intersection of with the bisector of and is a point on . Let the circumcenter of and , , . If is an equilateral triangle, prove that one of the triangles , , must be equilateral.

3. A slip on an integer is an operation that consists in choosing a prime divisor of and replacing by Starting with an arbitrary integer greater an equal to 5, we successively apply the slip operation on it. Show that one eventually reaches 5 no matter the slips applied.

ΗΜΕΡΑ 2η

4. Find all positive integers , , such that and are prime numbers and
5. If , , are positive numbers satisfying

Find all the possible values of .
6. Let be an acute triangle and , , be the feet of the altitudes through , , respectively. Call and the feet of the perpendicular lines from and to and , respectively. Let be the symmetric of with respect to and be the symmetric of with respect to . If , prove that