Turkey Team Selection Test 2014

Ημέρα 1η 

1. Find the number of permutations of the such that, for all , . 

2. Find all functions from real numbers to itself such that for all real numbers the equation

 

holds. 

3. Let and be the radii of the incircle, circumcircle and A-excircle of the triangle with , respectively. and are the centers of these circles, respectively. Let incircle touches the at , for a point the condition holds.

Prove that 

Ημέρα 2η 

1. Find all pairs of positive odd integers, such that

  and . 

2. A circle cuts the sides of the triangle at and ; and ; and , respectively. Let be the center of . is the circumcenter of the triangle , is the circumcenter of the triangle , is the circumcenter of the triangle . Prove that and concur. 

3. Prove that for all all non-negative real numbers with

Ημέρα 3η 

1. Let be a point inside the acute triangle with . is the midpoint of the segment . and intersect at , and and intersect at . Prove that . 

2. , and for all the sequence defined as,

If a prime divides for a natural number n, prove that there is a integer such that

3. At the bottom-left corner of a chessboard, there are some green worms and at the top-left corner of the same chessboard, there are some brown worms. Green worms can move only to right and up, and brown worms can move only to right and down. After a while, the worms make some moves and all of the unit squares of the chessboard become occupied at least once throughout this process. Find the minimum total number of the worms.

Πηγή: artofproblemsolving