[/quote]
,
be a scalene triangle such that 
is its incircle. 
is tangent to 
.
)
.
,
,
be the incircles of the triangles 
,
,
,
is inscribed in a circle 
where 
,
is the intersection of diagonals 
and 
,
bisects the angle 
prove that 
be the set of all positive integers. Find all functions 
such that for any positive integers 
and 
,
,
,
,
are equal.
are internally tangent at point 
.
and 
and 
respectively. Rays 
.
.
and two fixed points 
on that circle, let 
be a moving point on 
be the midpoint of 
be the three heights of 
,
such that 
.
be the intersection of 
and 
,
be the second intersection of 
and 
.
always goes through a fixed point.
intersects 
.
,
such that 
.
be the center of 
.
always goes through a fixed point.
where 
where 
.
cannot be a perfect cube of a positive integer.
is given. Let 
segments of real line of type ![$[i, j]$](http://latex.artofproblemsolving.com/a/6/4/a644edd67caa59eaded765c417b6d9822f782ec2.png)
,
and 
are integers, 
.
is said to be valuable if the intersection of any two segments from 
is either empty, or is a segment of nonzero length belonging to 
![\[P(x) P(x+1) = P(x^2 + x + 1)\]](http://latex.artofproblemsolving.com/b/e/f/beff3420fa8394f4f98c031d81eea6a5a3b8d28c.png)
for all real numbers 
.
and![\[(x-y)[f(x)+f(y)]\leqslant f(x^2-y^2), \forall x,y \in \mathbb{R}\]](http://latex.artofproblemsolving.com/8/0/b/80bb1d7a992c9683a0c2a07b2febe6271f4e60f5.png)
such that 
,
and 
.
is prime, 
with 
and 
?
Y by
A (possibly non-convex) planar polygon P is good if no two sides of P are parallel.
For any good polygon P, we may take any three sides of P and extend them into lines. These lines
intersect to form a triangle. Such a triangle is called a peritriangle of P. Let f(P) denote the minimal
number of peritriangles of P whose union completely cover P.
For each positive integer n, find all possible values of f(P) as P ranges over all good n-gons.
For any good polygon P, we may take any three sides of P and extend them into lines. These lines
intersect to form a triangle. Such a triangle is called a peritriangle of P. Let f(P) denote the minimal
number of peritriangles of P whose union completely cover P.
For each positive integer n, find all possible values of f(P) as P ranges over all good n-gons.
