[/quote]
,
be a triangle, and let 
be a point on side 
.
at 
and ray 
at 
. The circumcircle of triangle 
intersects the circumcircle 
of triangle 
distinct from point 
.
and 
intersect 
and 
respectively.
such that 
for all real number 
and 
and 
on the plane such that 
is an equilateral . Let 
and 
at 
and 
at 
,
on 
satisfies 
and 
.
be the point where the perpendicular from 
to line 
meets the angle bisector of 
.
holds.
be a positive integer such that 
.
.
be lines in the plane such that no two of them are parallel and no three of them are concurrent. Let 
be the intersection point of lines 
.
on 
and 
on 
such that 
Interior points.
)
.
,
be the orthocenter of it and 
be any point on the side 
.
are on the segments 
,
and 
are cyclic. The segments 
and 
intersect at 
is a point on 
such that 
is tangent to the circumcircle of triangle 
at 
and 
intersect at 
.
and 
and 
then prove 
be the set of positive real numbers.
such that there exists a function 
with the property:![\[
f(y^2f(x) + y + c) = xf(x+y^2)
\]](http://latex.artofproblemsolving.com/e/3/0/e30ee6d9f69cc43422a5c2b9813192a632d359cb.png)
for all 
.
be non-negative real numbers such that 
Prove that
Y by aidan0626, GodGodGodGodGoose
Now unfortunately I don't know the difficulty of the problems posted here but I'll try to replicate:
Bob has 20 apples and 19 oranges. How many ways can he split the fruits between 7 people if each person must have at least 1 apple and 2 oranges?
After looking at the other posters I realized just how bashy this is
Also I can only edit this message for now since new AoPS users can only send 6 messages every day
Bob has 20 apples and 19 oranges. How many ways can he split the fruits between 7 people if each person must have at least 1 apple and 2 oranges?
After looking at the other posters I realized just how bashy this is
Also I can only edit this message for now since new AoPS users can only send 6 messages every day
This post has been edited 1 time. Last edited by bibidi_skibidi, Apr 15, 2025, 4:19 AM
ways to split the apples and 
ways to split the oranges, and you just multiply those two together to get the final answer.
is what I got
be the number of oranges each person gets, so 
then use stars and bars. similar process for apples
