▪International Zhautykov Olympiad 2012

Ημέρα 1η

 1. An acute triangle is given. Let be an arbitrary inner point of the side . Let and be foots of perpendiculars from to and , respectively. Let and be the orthocenters of triangles and , respectively. Prove that the area of the quadrilateral does not depend on the position of on . 

2. A set of (unit) squares of a is called convenient if each row and each column of the table contains at least two squares belonging to the set. For each determine the maximum for which there exists a convenient set of that becomes inconvenient when any of its squares is removed. 

3. Let be three polynomials with real coefficients such that

for all . Prove that

    or   

for all . 

Ημέρα 2η

1. Do there exist integers and a function satisfying simultaneously the following two conditions

 

for any ,  and ? 

2. Equilateral triangles and are drawn on the diagonals of a convex quadrilateral so that and are on the same side of , and and are on the same sides of . Find

  if . 

3. Find all integer solutions of the equation the equation

  .