AoPS Academy
be an acute-angled triangle with 
, let 
be its circumcentre. The line through 
perpendicular to 
intersects circle 
again at 
. The tangents at 
and 
of intersect at 
. 
intersects at 
. 
intersects 
at 
. Let 
be the midpoint of 
.
are concyclic.
be real numbers. Knowing that there are positive real numbers 
, different two by two, such that
calculate the value of
in terms of .
be real numbers.Prove that
be a positive integer, and consider an 
grid. A right-down path is a sequence of grid cells such that each cell is either one cell to the right of or one cell below the previous cell in the sequence. A right-up path is a sequence of grid cells such that each cell is either one cell to the right of or one cell above the previous cell in the sequence. grid cannot be partitioned into less than right-down or right-up paths. For example, the following partition of the 
grid uses 
paths.
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 are the roots of the cubic equation 
. The value of 
is ...
so that 
and 
is ...
, 
, 
, 
has an average of 
and a median of 
. The largest number of such data is ...
are integers greater than 
which are four consecutive quarters of an arithmetic row with 
. If 
and 
are squares of two consecutive natural numbers, then the smallest value of 
is ..., with 
, 
and 
. The points 
and 
lies on the line segment . with 
and 
. The measure of the angle 
is ...
meet 
for each natural number 
. The value of 
is ....
provided that the difference of any two numbers at least 
is ...
which satisfies 
are as many as ... with 
and 
. Point lies on the side so that 
. Suppose 
is a point on the side extension 
so that 
is perpendicular to . The point lies on the ray such that 
and 
. The large angle 
is ... consists of integers with the following properties: For every three different members of there are two of them whose sum is a member of . The largest value of is ....
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 so that the game never ends for any initial square selection is ....
students, 
students play cricket, 
students play hockey and 
students play basketball. 
students play both basketball and hockey, 
students play both cricket and basketball, 
students play both cricket and hockey, and 
students play all three: basketball, hockey, and cricket. Find the number of students who do not play any game.
such that 
divides 
.
, points , , and are the midpoints of sides , 
, and , respectively. Prove that the area of the triangle with side lengths 
, 
, and 
has area ![$\tfrac{3}{4}[ABC]$](http://latex.artofproblemsolving.com/9/8/a/98a118cb3595f9ea91f8b38b7293cc211f81295a.png)
.
be a permutation of 
such that there are exactly ordered pairs of integers 
satisfying 
and 
. How many possible exist?
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
be a trapezoid with parallel sides 
,
.
again at 
respectively. Prove that 
are concurrent.
gives that 
and 
are cyclic, so radax
,
are concyclic, respectively. This is proved by![\[
\measuredangle AWZ
=\measuredangle CWZ
=\measuredangle CDZ
=\measuredangle CDB
=\measuredangle ABD
\]](http://latex.artofproblemsolving.com/8/c/f/8cf6ee67dfd80862f1a5ff05b28afedf37f9ef2a.png)
and same for