[/quote]
,
,
.
intersects 
.
lies on the circumcircle of triangle 
.
marbles in 
colors such that there are exactly 
marbles of each color. She wants to arrange them on 
shelves, 
marbles on each shelf such that for any 
colors there is a shelf that has marbles of those colors.
be a rearrangement of the integers from 1 to 2025.
,
to be the number of integers 
such that 
and 
.
are given such that 
and 
for all 
satisfying this condition.
.
are the orthocenter and circumcenter. Let 
be a point on the same side of 
as 
,
is equilateral. Let 
as 
is equilateral. Let 
as 
is equilateral. Let 
.
are collinear.
Prove that
Let 
Prove that
,![\[\left(1+\frac{a}{b}\right)\left(1+\frac{a}{b}+\frac{b}{c}\right)\left(1+\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)\geq 16\]](http://latex.artofproblemsolving.com/c/1/7/c17cc42f0feafca5ecbeacde7076daa5b2615c60.png)
Prove that
Where 
satisfying
Prove
,
,
,
.
such that for every 
![\[
f_n(x) = \frac{x}{1 + nx}, \quad \text{if and only if } f(x) = \frac{x}{1 + x}, \quad \forall x \geq 0.
\]](http://latex.artofproblemsolving.com/0/f/c/0fc36d9264eb7e103128c489aeae521a859c1fd4.png)
(For 
represents 
.
in the plane.Let 
be reflections of 
across the line 
be the midpoints of 
,
be the reflections of 
and 
Prove that
Where 
be the set of positive integers. Find all functions 
such that![\[f(m)+n \mid m^2f(m) - f(n) \]](http://latex.artofproblemsolving.com/8/5/5/85515a57bf60498134ceb168039fe734a5ff91a5.png)
for all positive integers 
and 
.
.
,
be its circumcentre and orthocentre respectively. Points 
lie on segments 
respectively, such that ![\[\angle ZOB=\angle YOC = 90^{\circ}.\]](http://latex.artofproblemsolving.com/8/f/e/8fe34cc0f04ff9a48f1a7eee89fc50d57272d936.png)
The perpendicular line from 
to line 
meets lines 
and 
at 
respectively. Let the tangents to the circumcircle of 
at points 
and 
meet at point 
.
are concyclic.
Y by NO_SQUARES, ehuseyinyigit
Let 
be a triangle with incircle 
, which touches sides 
and 
at points 
and 
, respectively. Choose points 
and 
on the line 
such that 
and 
. Let 
be the intersection of lines 
and 
. Define 
and 
as the intersections of 
and 
with lines 
and 
, respectively. Assume that 
is the intersection of 
and 
. Prove that 
.
be a triangle with incircle 
, which touches sides 
and 
at points 
and 
, respectively. Choose points 
and 
on the line 
such that 
and 
. Let 
be the intersection of lines 
and 
. Define 
and 
as the intersections of 
and 
with lines 
and 
, respectively. Assume that 
is the intersection of 
and 
. Prove that 
.Attachments:
at 
ia a parallelogram. 
intersect 
at 
intersect 
at 
.
bisect 
.
bisects segment 
,
