[/quote]
,
Write only the answers to the questions given.
to be more exact:
in years 2002-08 time was 90' for part A and 120' for part B
red balls and 
white balls. Audi took two balls at once from inside the bag. The chance of taking two balls of the same color is ...
unit. The area of the hexagon is ...
and 
.
is ...
so that 
and 
is ...
,
,
,
has an average of 
and a median of 
.
are integers greater than 
which are four consecutive quarters of an arithmetic row with 
.
and 
are squares of two consecutive natural numbers, then the smallest value of 
is ...
,
and 
.
and 
lies on the line segment 
.
and 
.
is ...
meet 
for each natural number 
.
is ....
provided that the difference of any two numbers at least 
is ...
which satisfies 
are as many as ...
and 
.
.
is a point on the side extension 
is perpendicular to 
and 
.
is ...
consists of 
with 
positive reals is ....
with 
is ....
square units. Two plots are said to be neighbors if they both have one side in common. Initially, there are a total of 
coins on the chessboard where each coin is only loaded exactly on one square and each square can contain coins or blanks. At each turn. You must select exactly one plot that holds the minimum number of coins in the number of neighbors of the plot and then you must give exactly one coin to each neighbor of the selected plot. The game ends if you are no longer able to select squares with the intended conditions. The smallest number of 
is almost squarefree if there exists a prime number 
such that 
and 
is squarefree. Prove that for any almost squarefree positive integer 
is an integer.
be a family of at most 
with the following property: we can consider 
and draw segments connecting these points such that points 
and 
are connected if and only if 
belongs to 
Determine the maximal value that the sum of the cardinalities of the subsets in 
can take.
?
such that![\[ f(x)f\left(x + 4f(y)\right) = xf\left(x + 3y\right) + f(x)f(y) \]](http://latex.artofproblemsolving.com/d/b/1/db14b64aa71cb3c8dbf53707f719f5d9a43bd7e4.png)
for any positive real numbers 
.
and for any non-empty subset 
she computes the sum ![\[s_A=\sum_{k
\in A}a_k.\]](http://latex.artofproblemsolving.com/7/8/a/78afe6d1036a006b2252023d8f9619285fe0c6d5.png)
She orders these sums 
Prove that there exists a subset 
with 
elements such that, regardless of the integers 
with 
and the circle 
of radius 
centered at 
Let 
be the midpoint of 
The line 
intersects 
Let 
Let 
be the intersection of 
and 
and 
.
,
be non-negative real numbers, no two of which are zero. Prove that :
with 
.
points on sides 
,
are angle bisectors on 
is right-angled isosceles.
be a positive root of the equation 
.
Y by ImSh95
Let 
be an acute-angled triangle and 
be a point in its interior. Let 
and 
be the images of under reflection in the sides 
, and 
, respectively. If is the orthocentre of the triangle 
and if the largest angle of the triangle that can be formed by the line segments
. and 
is 
, determine the value of 
.
be an acute-angled triangle and 
be a point in its interior. Let 
and 
be the images of under reflection in the sides 
, and 
, respectively. If is the orthocentre of the triangle 
and if the largest angle of the triangle that can be formed by the line segments
. and 
is 
, determine the value of 
.
is equilateral.
be 
and 
midpoint of 
to 
on 
,
.
is the inscribed triangle of 
centered around this centroid. This sends 
,
about the centroid sends the orthocenter to the circumcenter (euler line), so 
is the midpoint of 
is perpendicular to 
is the perpendicular bisector of 
then the circumcircles of 
and 
have equal radii which can be shown with Sine Rule easily. There is an inversive proof too, which I don't know, but I am pretty sure it exists.
.
lies on perpendicular bisector of 
,
.
.
.
