be vertices of regular pentagon inscribed in a circle whose radius is 
and center is at 
.
are integers whose greatest common divisor is 1. Let 
be a set of integers with the following properties:
,
.
(not necessarily distinct), 
.
,
,
.
.
,
such that 
is divisible by 
,
.
be a quadrilateral circumscribed about a circle, whose interior and exterior angles are at least 60 degrees. Prove that![\[
\frac{1}{3}|AB^3 - AD^3| \le |BC^3 - CD^3| \le 3|AB^3 - AD^3|.
\]](http://latex.artofproblemsolving.com/5/5/e/55ee0fba8392ee81b7b37da45d99755ea6d6754a.png)
such that 
for all 
with parallel sides 
and 
,
on line 
outside segment 
inside segment 
such that 
.
the point of intersection of 
,
the point of intersection of 
be the midpoint of segment 
if and only if 
.
with an area of 
,
.
,
.
.
Y by baopbc, doxuanlong15052000, Tawan, Adventure10, Mango247, Rounak_iitr
Let 
be an arbitrary triangle and 
is the circumcenter of 
.Points 
lie on 
,respectively such that the reflection of 
WRT 
is tangent to circumcircle of 
.Prove that the circumcircle of triangle 
is tangent to circumcircle of triangle 
.
be an arbitrary triangle and 
is the circumcenter of 
.Points 
lie on 
,respectively such that the reflection of 
WRT 
is tangent to circumcircle of 
.Prove that the circumcircle of triangle 
is tangent to circumcircle of triangle 
.
Let 
be the symmetric of 
through 
.
be the intersection of 
and 
.
through 
which gives 
are concyclic.
:
and 
are tangent at 
if and if only:
![[asy]
import graph; size(12.565527202436744cm);
real labelscalefactor = 0.5; /* changes label-to-point distance */
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */
pen dotstyle = black; /* point style */
real xmin = -3.556734891924745, xmax = 20.008792310512, ymin = -8.60342049983383, ymax = 13.620567560036616; /* image dimensions */
/* draw figures */
draw(circle((9.184355987566615,9.340643498633334), 4.136210591851162));
draw(circle((10.806120374183049,-1.159828565994827), 6.488761563676291));
draw((4.388591091050947,-2.118653122618589)--(14.53262231503499,4.152158813044161));
draw((14.53262231503499,4.152158813044161)--(9.81569496288056,5.2528997250595415));
draw((9.81569496288056,5.2528997250595415)--(4.388591091050947,-2.118653122618589));
draw((9.81569496288056,5.2528997250595415)--(13.053525816289774,7.878534440922462));
draw((9.81569496288056,5.2528997250595415)--(5.9748832427064436,11.949766485412887));
draw((5.9748832427064436,11.949766485412887)--(13.053525816289774,7.878534440922462));
/* dots and labels */
dot((5.9748832427064436,11.949766485412887),linewidth(3.pt) + dotstyle);
label("$A$", (5.353109979458428,12.279028417470405), NE * labelscalefactor);
dot((13.053525816289774,7.878534440922462),linewidth(3.pt) + dotstyle);
label("$B$", (13.554217190617939,7.8241059817788345), NE * labelscalefactor);
dot((9.81569496288056,5.2528997250595415),linewidth(3.pt) + dotstyle);
label("$X$", (8.517117391171203,5.748517119695262), NE * labelscalefactor);
dot((14.53262231503499,4.152158813044161),linewidth(3.pt) + dotstyle);
label("$D$", (14.971692511065262,4.356353858541647), NE * labelscalefactor);
dot((4.388591091050947,-2.118653122618589),linewidth(3.pt) + dotstyle);
label("$C$", (3.7584502439551897,-2.4019659728768143), NE * labelscalefactor);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
[/asy]](http://latex.artofproblemsolving.com/c/f/0/cf0d455a8ec1b0d2cf91004a5210ffff89e2003d.png)
over 
touches 
at 
be the Miquel point of 
w.r.t. 
.
,
,
by simmetry, therefore: 
as claimed. 
hence, 
and 
are tangent to each other at 
By the Miquel 
concur at some point 
Hence 
tangent at 
in 
and 
are concyclic, we are just directly done by inversion at 
swapping 
and 
,
In that case, 
So this motivates us to define 
