2009 Zhautykov International Olympiad Problems
Contents
Day 1
Problem 1
Find all pairs of integers , such that
![$x^2 - 2009y + 2y^2 = 0$](http://latex.artofproblemsolving.com/3/4/5/34504ec6abb78e0f5a2a37464f0303dd348db61f.png)
Problem 2
Find all real , such that there exist a function
satisfying the following inequality:
![$x + af(y)\leq y + f(f(x))$](http://latex.artofproblemsolving.com/6/9/1/6917ef956b9cfc0828cfc34250b25b397afcaf72.png)
for all .
Problem 3
For a convex hexagon with an area
, prove that:
![$AC\cdot(BD + BF - DF) + CE\cdot(BD + DF - BF) + AE\cdot(BF + DF - BD)\geq 2\sqrt {3}$](http://latex.artofproblemsolving.com/d/3/4/d34e299c8bf6393bbc5553f97c9a44c6ee23bcb0.png)
Day 2
Problem 4
On the plane, a Cartesian coordinate system is chosen. Given points on the parabola
, and points
on the parabola
. Points
are concyclic, and points
and
have equal abscissas for each
.
Prove that points
are also concyclic.
Problem 5
Given a quadrilateral with
. Point
is chosen on segment
so taht
. Rays
and
intersect at point
. Points
and
are feet of perpendiculars from points
and
to lines
and
, respectively.
Prove that
.
Problem 6
In a checked table,
squares are colored in black. We call a line any of rows, columns, or any of two diagonals of the table. In one step, if at least
of the squares in some line are black, then one can paint all the squares of this line in black. Find the minimal value of
such that for some initial arrangement of
black squares one can paint all squares of the table in black in some steps.