[/quote]
,
such that 
divides 
for any positive integers 
and 
.
such that for all 
,
.
with 
and incenter 
.
be the mixtilinear circle at vertex 
,
and also tangent to lines 
and 
.
passes through points 
and 
and is tangent to 
,
and 
.
,
are chosen independently and uniformly at random on a circle of circumference 
.
,
,
of length 
on the circle, we have 
.
on the circle that contains all the points 
is a pair of real numbers such that
.
for which the sequence of real numbers with 
for all 
is periodic from the beginning.
as well as all values of 
which attain it.
and a constant 
,
and 
are also friendly polynomials. Prove that 
.
satisfies the following condition.
,![$\sum_{k=1}^{n}\frac{1}{2}\left(1 - (-1)^{\left[\frac{n}{k}\right]}\right)a_k=1$](http://latex.artofproblemsolving.com/3/1/0/3109e3f024dc59c2cdad60efa270494d6095c330.png)
holds.
,
.
be reals such that 
and 
Prove that
grid of rooms, each initially at a temperature of 
C
C
degrees. The ice spell lasts for 
nanoseconds, after which temperatures revert to their initial values.
satisfy both of the following conditions.
,
,
.
that satisfies the equation 
and 
.
,
be real numbers such that 
.
is the interval ![$\left[-\frac{2}{3}, \frac{2}{3}\right]$](http://latex.artofproblemsolving.com/4/f/2/4f22df2b391cf34b9b4f86634575e3e0ad23a386.png)
.
selects a graph with 
vertices such that each vertex is incident to at least one edge. Then 
deletes some of the edges (possibly none) from the chosen graph. Finally, 
Y by Adventure10, Mango247, and 1 other user
Let 
be a triangle such that 
,
the points
and 
are tangency points for the ex-circles relative to sides 
and 
,
respectively, and
the incircle.
The line
intersects the cincumcircle of the triangle in the point 
.
The line that trough that is perpendicular to 
, intersects the line 
in 
.
Prove that the lines
and 
are parallels.
be a triangle such that 
,the points
and 
are tangency points for the ex-circles relative to sides 
and 
,respectively, and
the incircle.The line
intersects the cincumcircle of the triangle in the point 
.The line that trough
that is perpendicular to 
, intersects the line 
in 
.Prove that the lines
and 
are parallels.
let 
be foot of the angle bisector 
Let parallel to CK through 
its area.
cut by the line 
In addition, 
hence 
Since 
and 
we have 
and 
are similar, hence 
and
^{2}+[c^{2}-(b-a)^{2}](b+a)^{2}}{4s(s-c)}\cdot \frac{c}{2(b^{2}-a^{2})}=$](http://latex.artofproblemsolving.com/9/f/0/9f06e648e37c31283a72838e9e389ee7cf4de792.png)
are centrally similar by SAS, hence 
with circumcircle 
and incircle 
.
and 
so that 
. Prove that 
.
.![$[CC']$](http://latex.artofproblemsolving.com/a/8/1/a81ee27e589c3262d8fb37e72a84a363bca0eb34.png)
,![$[PP']$](http://latex.artofproblemsolving.com/c/8/5/c851691a4111d3eed5ac07f9d7ca3c25d061d0bf.png)
of 
and 
.
At first I"ll find the position of 
.
is harmonically, i.e. 
.
At second I"ll find the positions of 
,
and I"ll value the ratio 
.
.
.
.
touch 
at 
.
meets 
at 
meet 
and 
so 
are rectangle so 
and 
and 
are congruent.
.
and 
are similar.
and 
are similar
are cyclic so
be a triangle such that 
the points 
and 
are tangency points for the ex-circles relative to sides 
and 
respectively and 
the incircle. The line 
intersects the cincumcircle of the triangle 
The line that goes trough 
perpendicular to 
intersects the line 
in 
Prove that 
and 
are parallel.
Then
has infinite point 
passing through 
hence they have the same infinite point 
Its equation is then
Coordinates of its infinite point 
are:
and obviously we have 
l
and define similarly 
.
and 
.
and 
we get the collinearity of 
.
.
for proving 
.
and similarly for 
.
which is obvious since 
and similarly for 
.
be the projection of 
on 
.
and 
be the projection of 
,
and 
,
and 
are homothety (also congruent) .
are concyclic at 
,
,
and 
.
,
. i.e. 
be the tangent point of 
with 
.
and parallel to 
intersect 
at 
,
are concyclic . ... 
are concyclic (Reim theorem). ... 
(Reim theorem). i.e. 
at 
respectively, incircle touches 
.
.
concyclic, so 
,
.
,
.
be the intersection point of 
be the midpoint of 
.
,
,
is perpendicular to 
and 
,
be the midpoint of the arc 
of the circumcircle of 
be its circumcenter. Let 
and 
be the reflection points of 
and 
is perpendicular to 
.
and 
are perpendicular to 
and 
.
is concyclic with the center 
,
and 
is concyclic. So 
is concyclic with the center 
