▪ China National Olympiad 2012

ΗΜΕΡΑ 1η

1. In the triangle , is biggest. On the circumcircle of , let be the midpoint of and be the midpoint of . The circle passes through and is tangent to at , the circle passes through and is tangent at . and intersect at and . Prove that bisects .

2 Let be a prime. We arrange the numbers in as a matrix . Next we can select any row or column and add to every number in it, or subtract from every number in it. We call the arrangement good if we can change every number of the matrix to in a finite number of such moves. How many good arrangements are there? 

3 Prove for any , there exists an increasing sequence of positive integers satisfying:

1) for any ; 

2) There exists a positive integer and , satisfying if and only if .

ΗΜΕΡΑ 2η

1. Let where . For any reals satisfying , find the maximum of . 

2. Consider a square-free even integer and a prime , such that

1) ;

2) ;

3) There exists an integer such that .

Prove that there exists pairwise distinct positive integers such that . 

3. Find the smallest positive integer such that, for any subset of with , there exist three elements in such that , , , where are in and are distinct integers.