is the point of tangency of the incircle with the side 
and 
are the intersections of the angle bisectors of 
and 
with the circumcircle of 
be the antipodal point of 
in the circumcircle of 
,
be the antipodal point of 
.
and 
are similar.
passes through 
and 
again at points 
respectively, and intersects segment 
and 
such that 
and 
at 
at 
.
is parallel to 
with real coefficients such that 
.
polynomial:![\[\underbrace{P(P(\ldots P(P}_{n}(x))\ldots))- \underbrace{Q(Q(\ldots Q(Q}_{n}(x))\ldots))\]](http://latex.artofproblemsolving.com/5/e/1/5e17b712fca5c08dc9004701f8fc4c12fb81b204.png)
.
Prove that
Let 
Prove that![$$ (a+b)(a+1)(b+1) \leq \frac{3}{2}+\sqrt[3]{6}+\sqrt[3]{36}$$](http://latex.artofproblemsolving.com/f/5/e/f5e68204507aa9c3e71c28844bbee3a804546b94.png)
.
be the midpoint of 
be an arbitrary point on 
.
over 
,
and 
lies on 
.
define 
if 
or 
,
where 
are all prime factors of 
with nonnegative integer coefficients such that 
divides 
for every nonnegative integer 
satisfy the equation 
.
such that
be a given integer. Prove that infinitely many terms of the sequence 
,![\[a_k=\left\lfloor\frac{n^k}{k}\right\rfloor,\]](http://latex.artofproblemsolving.com/8/3/2/832113b1ef1fe211e1eaead25129873aead1dab4.png)
are odd. (For a real number 
,
denotes the largest integer not exceeding 
all of whose digits are odd.
,
,
,
be distinct positive integers, 
.
and 
such that 
does not divide any of the numbers 
,
,
.
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 
.
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 
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 
and 
,
In that case, 
So this motivates us to define 
