[/quote]
,
,
,
,
,
,
initially contains one coin. The following operations are allowed
,
,
;
,
,
and 
.
coins.
be a real number and let 
be arbitrary real numbers in 
.
and 
are then defined recursively by
.
for all sufficiently large 
.
is a prime number
.
denote its incenter and 
its 
denote the midpoint of arc 
.
meets 
.
at 
and 
meets 
a second time at 
,
meets 
a second time at 
.
passes through the antipodal to 
be a triangle with circumcircle 
intersects the lines 
,
at points 
,
,
of the segments 
.
be a positive integer. Prove that there are infinitely many primes, each of which divides at least one term of the sequence 
,
.
![$\mathbb{Z}[x]$](http://latex.artofproblemsolving.com/4/b/7/4b7d768fffbb7a1a6163af392ed348f2d49d8783.png)
denote the set of single-variable polynomials in ![$\theta : \mathbb{Z}[x] \to \mathbb{Z}[x]$](http://latex.artofproblemsolving.com/0/3/2/032a637d4ae21da2c0ddd8dc127723807f11cc5f.png)
(i.e. functions taking polynomials to polynomials)![$p, q \in \mathbb{Z}[x]$](http://latex.artofproblemsolving.com/7/e/7/7e7eb925258a33e456ef4ff722c1f943dc155d10.png)
,
;![$p \in \mathbb{Z}[x]$](http://latex.artofproblemsolving.com/0/2/6/0267b98c16b400dda1f081877e9b84fe672ce0e7.png)
,
has an integer root if and only if 
does.
of positive real numbers that satisfy the system:![\[
\begin{aligned}
11bc - 36b - 15c &= abc \\
12ca - 10c - 28a &= abc \\
13ab - 21a - 6b &= abc.
\end{aligned}
\]](http://latex.artofproblemsolving.com/1/7/0/170e1cfaee90f5c37428fa75192ac889261a8f83.png)
that verify the following equation for all 
:
is a function for which the expression 
for all 
is always a perfect square. Prove that 
for all 
.
such that 
.
be a quadrilateral inscribed in a circle 
,
and 
are tangent to 
.
and 
and 
the reflections of 
,
be the midpoint of 
and 
are perpendicular.
.
be the feet of the altitudes from 
to 
respectively. Let 
be the circle with diameter 
be the intersection of 
and 
be the intersection of 
be the lines through 
respectively. Let 
and 
.
and 
.
be the line through 
and let 
be the reflection of 
is tangent to 
.
Y by centslordm
The incircle of 
tangents 
, 
, 
at 
, 
, 
, respectively. The projections of 
, 
to 
are 
, 
, respectively; the projections of , 
to 
are 
, 
, respectively; and the projections of , to 
are 
, 
, respectively. Show that the circumcircle of the triangle formed by 
, 
, 
is tangent to the incircle of .
tangents 
, 
, 
at 
, 
, 
, respectively. The projections of 
, 
to 
are 
, 
, respectively; the projections of , 
to 
are 
, 
, respectively; and the projections of , to 
are 
, 
, respectively. Show that the circumcircle of the triangle formed by 
, 
, 
is tangent to the incircle of .
However, my friend told me it has been found by @KevinCheng 
But I still would like to state my proof below.
are 
are 
are 
.
be the rectangular hyperbola passing through 
be the orthocenter and the orthic triangle of 
is the pedal triangle 
of 
denote the Simson line of 
WRT 
.
,
are the projections of 
.
,
since 
is a self-polar triangle WRT 
.
.
