[/quote]
,
be an isosceles trapezoid with 
.
be the midpoint of 
.
and 
the circumcircles of the triangles 
and 
,
be the crossing point of the tangent to 
with the tangent to 
.
is tangent to 
is a positive integer. Call all positive divisors of 
and 
beautiful divisors and difference of any 
we have 
.
.
,
,
,
,
meet the sides 
,
,
at points 
,
,
,
be the incenter of triangle 
,
be the foot of the perpendicular from 
.
is cyclic, then the center of its circumcircle lies on segment 
.
.
intersects 
at 
,
at 
passes from circumcenter of 
.
be an infinite sequence of real numbers in the interval 
.![\[ \frac{a_1}{1},\frac{a_2}{2},\frac{a_3}{3},\frac{a_4}{4},\ldots.\]](http://latex.artofproblemsolving.com/3/c/e/3cef10f63e9c2dde90b29c742ece8fd31e0a8e94.png)
with rational coordinates lie on a plane. It turned out that the distance between every pair of points is an integer. Prove that there exist points 
with integer coordinates such that 
for every pair 
and 
,
and 
.
and internally tangent to 
.
and 
intersect at a point lying on 
be inscribed in the circle 
.
intersects 
at points 
,
be the reflection of 
of triangle 
lies on the circle 
and 
coincide.
on the board. For each procces, the stutent deletes the number x and replaces it with either 
or 
or 
.
on the board?
satisfying
Prove
:![\[
1 \cdot 3 \cdot 5 \cdot \ldots \cdot (2n + 1) \mid (4n + 3)(4n + 5) \cdot \ldots \cdot (8n + 3).
\]](http://latex.artofproblemsolving.com/3/1/a/31a0eac3c7ca298c8fa221083dc94e127e705784.png)
and 
.
?
and 
.
.
.
Y by CZRorz, Adventure10, Mango247
Let 
be a chord of a circle 
centered at 
, and let 
be a point on the segment such that 
. Two circles through , internally tangent to at 
and 
, respectively, meet again at 
. Let 
be one of the points of intersection of the line 
and the circle , and let the lines and 
meet at 
. Prove that the line 
is tangent to the circle .
Theoklitos Paragyiou (Cyprus)
be a chord of a circle 
centered at 
, and let 
be a point on the segment such that 
. Two circles through , internally tangent to at 
and 
, respectively, meet again at 
. Let 
be one of the points of intersection of the line 
and the circle , and let the lines and 
meet at 
. Prove that the line 
is tangent to the circle .Theoklitos Paragyiou (Cyprus)
This post has been edited 1 time. Last edited by parmenides51, Jul 20, 2020, 4:54 AM
Reason: latex
Reason: latex
and 
,
and 
are tangent to 
it's so easy note that: 
is cyclic, by simple angle chasing we get 
and 
are the internal bisector and external bisector of 
respectively.
and 
,
is the polar of 
is tangent to 
