[/quote]
,
Prove that: 
is the sum of 
square numbers
Let 
and 
is the sum of 
is the sum of 
and incenter 
and let 
be the midpoint of 
.
,
such that 
,
,
.
intersects 
other than 
and 
meet on 
,
,
,
,
,
initially contains one coin. The following operations are allowed
,
,
;
,
,
and 
.
coins.
Prove that
its orthocenter, 
its circumcenter, and 
its circumradius. Let 
,
,
.
.
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 
.
its 
denote the midpoint of arc 
.
meets 
.
at 
and 
meets 
a second time at 
,
meets 
a second time at 
.
passes through the antipodal to 
,
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)
Y by
The two circles 
and 
intersect at points 
and 
. The line 
passes through point and intersects the circles and at points 
and 
, respectively. Let 
and 
be the orthocenters of triangles 
and 
, respectively, and let 
and 
be points diametrically opposed to in circles and , respectively. The lines 
,
intersect at the point 
, and the lines 
and 
intersect at the point 
. Prove that the midpoint of the segment 
lies on the line .
and 
intersect at points 
and 
. The line 
passes through point and intersects the circles and at points 
and 
, respectively. Let 
and 
be the orthocenters of triangles 
and 
, respectively, and let 
and 
be points diametrically opposed to in circles and , respectively. The lines 
,
intersect at the point 
, and the lines 
and 
intersect at the point 
. Prove that the midpoint of the segment 
lies on the line .This post has been edited 1 time. Last edited by parmenides51, Jun 21, 2021, 10:47 AM
