[/quote]
,
and circumcentre 
.
and 
lie on 
and 
.
meet 
at 
,
meet 
.
and 
have an intersection lie on line 
.
be a positive integer. Alex plays on a row of 9 squares as follows. Initially, all squares are empty. In each turn, Alex must perform exactly one of the following moves:
Choose a number of the form 
,
a non-negative integer, and place it in an empty square.
Choose two (not necessarily consecutive) squares containing the same number, say 
and erase the number in the other square.
,
and![\[ f(x+y)+f(x^2+f(y))=f(f(x))^2+f(x)+f(y)+y,\forall x,y\in\mathbb{R}\]](http://latex.artofproblemsolving.com/c/8/9/c895ae7fdf8d7f284ac9fc94cc077d6edad6cbf0.png)
be a convex hexagon with 
and 
.
and 
.
and 
are the centers of mass of 
and 
,
.
be a point in the plane, distinct from the vertices of 
.
the reflections of 
and 
,
are collinear if and only if 
coincide, prove that 
given that there exist positive integers 
satisfying
and 
as follows:
where 
are all the positive divisors of 
.
Find all positive integers 
.
Find all positive integers 
.
be an odd prime number. How many 
of 
are there, the sum of whose elements is divisible by 
be real numbers satisfying
be a fixed square and 
a variable point on segment 
.
is constructed such that 
is on segment 
and 
.
and 
.
board, two opposite sides have been joined, forming a cylinder. Determine whether it is possible to place 
.
.
denote the number of positive integers less than 
has at least 
solutions in 
that can be written in the form
where 
are pairwise coprime positive integers such that 
,
,
.
Y by
In a triangle 
with right angle 
, the altitude 
is drawn. An arbitrary circle passing through points and 
meets the segments 
, 
and 
for the second time at points 
, 
and 
respectively. Segments 
and 
meet at point 
. What is greater: the area of triangle 
or the sum of areas of triangles 
and 
? (5 marks)
with right angle 
, the altitude 
is drawn. An arbitrary circle passing through points and 
meets the segments 
, 
and 
for the second time at points 
, 
and 
respectively. Segments 
and 
meet at point 
. What is greater: the area of triangle 
or the sum of areas of triangles 
and 
? (5 marks)Attachments:
This post has been edited 1 time. Last edited by QueenArwen, Mar 24, 2025, 12:25 PM
![\[A(CPT)-A(THR)=A(CPR)-A(HPR)=PR \cdot \frac{h_C-h_P}{2}=CQ \cdot \frac{h}{2}\]](http://latex.artofproblemsolving.com/d/6/c/d6c6a95a33ebc9489d296c329401dd10e348ca92.png)
where 
and 
are the altitudes from 
,
is the altitude from 
.
and 
.
to both sides. We are actually asked which has a greater area, triangle 
or quadrilateral 
.
is a diameter as well so we might want to introduce 
,
,
.
and 
.
,
and 
are parallelograms, we have the areas of the same color are equal, hence the of red area and the blue one (in the previous image) are equal.
