[/quote]
,
Prove that
,
,
,
,
,
initially contains one coin. The following operations are allowed
,
,
;
,
,
and 
.
coins.
be a triangle, 
its orthocenter, 
its circumcenter, and 
its circumradius. Let 
be the reflection of the point 
across the line 
,
be the reflection of the point 
across the line 
,
be the reflection of the point 
across the line 
.
.
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 
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 
.
Y by
A circle 
is given. Circles 
, 
, 
, 
, 
and 
touch internally. In addition, the circles , and touch each other externally, the circle touches the circles and externally, the circle touches the circles and externally, the circle touches the circles and externally (see fig.). Prove that there is a circle tangent to the common internal tangent pairs of circles 
and 
, 
(we assume 
).
is given. Circles 
, 
, 
, 
, 
and 
touch internally. In addition, the circles , and touch each other externally, the circle touches the circles and externally, the circle touches the circles and externally, the circle touches the circles and externally (see fig.). Prove that there is a circle tangent to the common internal tangent pairs of circles 
and 
, 
(we assume 
).This post has been edited 3 times. Last edited by parmenides51, Apr 5, 2021, 2:23 PM
