[/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
is inscribed in circle 
.
denote its incenter and 
its 
denote the midpoint of arc 
.
meets 
.
at 
and 
respectively. The circumcircle of 
meets 
a second time at 
,
meets 
a second time at 
.
passes through the antipodal to 
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 
and 
the reflections of 
with 
,
and 
are perpendicular.
be the feet of the altitudes from 
to 
respectively. Let 
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
Around triangle 
, a circle 
is circumscribed with center at point 
. Circle 
touches line 
at point 
and passes through point 
, and circle 
touches line 
at point and passes through point 
. Through point , a line is drawn that intersects circle for second time at point 
and circle for second time at point 
. Point 
is the midpoint of segment 
. Find the angle 
.
, a circle 
is circumscribed with center at point 
. Circle 
touches line 
at point 
and passes through point 
, and circle 
touches line 
at point and passes through point 
. Through point , a line is drawn that intersects circle for second time at point 
and circle for second time at point 
. Point 
is the midpoint of segment 
. Find the angle 
.This post has been edited 1 time. Last edited by parmenides51, Feb 16, 2021, 11:14 AM
