[/quote]
,
be a triangle with 
-
,
and 
to 
and 
meet the circumcircle again at 
and 
,
and 
meet 
again at 
and 
.
is tangent to the circumcircle.
and a point 
outside the circle such that 
. The perpendicular bisector of 
at 
and 
.The segment 
intersects 
at 
intersects 
at 
.The rays 
intersect 
respectively. Prove that the triangles 
and 
are equilateral triangles and the lines , 
are concurrent.
such that for any real numbers 
and 
,![\[
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\]](http://latex.artofproblemsolving.com/8/4/3/8437449cd3f8f0da24d17a60b55dcb563e6b5655.png)
of positive integers satsifying 
.
penguin and each one is numbered using numbers 
.
is called 
if 
where 
is 
prime divisor of 
)
where 
is number of 
be real numbers so that 
and 
Show that
of positive integers satisfying the following two conditions:
,
is in 
,
and 
;
such that 
is not divisible by 
.
be a scalene acute triangle with incentre 
and circumcentre 
.
cross 
.
,
cross 
at 
and let 
cross 
at 
.
and 
are concurrent on the external bisector of 
.
with integer coefficients, satisfying 
for all non-negative integers 
positive real numbers such that 
. Prove that :
be a non-constant monic polynomial with integer coefficients and let 
be an infinite sequence. Prove that there are infinitely many primes, each of which divides at least one term of the sequence 
.
be an integer, and 
can divide 
let 
be all distinct prime divisors of 
is an integer. ( where 
is defined as the number of positive integers 
that are relatively prime to 
be real numbers . Prove that
Where 
?
Y by buratinogigle, mineiraojose, AmirKhusrau, enhanced, Adventure10, Mango247, GeoKing
Given a 
with incenter 
circumcenter 
Let 
be the projection of 
on 
be the reflection of in and 
be points on the tangent 
of 
at 
If 
is the second intersection of 
with 
respectively. Prove that 
where 
is the radius of 
P.S.This problem is a generalization of 2014 Taiwan TST 2 Quiz 2 Problem 1
with incenter 
circumcenter 
Let 
be the projection of 
on 
be the reflection of in and 
be points on the tangent 
of 
at 
If 
is the second intersection of 
with 
respectively. Prove that 
where 
is the radius of 
P.S.This problem is a generalization of 2014 Taiwan TST 2 Quiz 2 Problem 1
This post has been edited 2 times. Last edited by TelvCohl, Apr 3, 2020, 9:12 AM
Reason: Make the statement more clear
Reason: Make the statement more clear
are projections of 
.
cuts 
again at 
.
is the line passing through 
is circumcenter of 
are two points on line 
cut circumcircle of triangle 
agian at 
then 
.
denote the A-excenter of 
at 
respectively and cuts 
again at 
is the circumcenter of 
lying on perpendicular 
from 
to 
(antiparallel to 
WRT 
)
cut by the line 
we get
is a perspectivity between 
and 
Recall that a general perspectivity would be 
for 
Hence all lines 
go through a fixed point. When 
When 
we have then
are collinear 
of bigger arcs 
of 
and also goes through the midpoints of some arcs of 
.
and on 
be the intersection of 
with 
respectively and 
be the point such that 
then 
so by Reim's theorem we get 
and the second intersection 
of 
with 
lie on 
hence (by Reim's theorem again) we conclude that 
Similarly, if 
is the point such that 
then 
lies on 
so 
