[/quote]
,
.
,
the midpoints of 
,
.
with diameter 
,
at a point 
.
meets line 
at 
and 
intersect at a point 
on 
.
Prove that: 
is the sum of 
square numbers
Let 
and 
is the sum of 
is the sum of 
and incenter 
and let 
,
are selected on sides 
,
such that 
,
,
.
intersects 
other than 
.
and 
meet on 
,
,
,
,
,
initially contains one coin. The following operations are allowed
,
,
;
,
,
and 
.
coins.
Prove that
across the line 
,
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 
.
its 
.
meets 
at 
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.
Y by
Let 
be the point of intersection of the altitudes of the triangle 
. Let the circle passing through points 
have center 
and intersect the segment 
at the point 
(
). Let 
be the intersection point of the line 
with segment 
. Let 
be the center of the circumscribed circle of the triangle 
. Prove that the points 
lie on a circle.
be the point of intersection of the altitudes of the triangle 
. Let the circle passing through points 
have center 
and intersect the segment 
at the point 
(
). Let 
be the intersection point of the line 
with segment 
. Let 
be the center of the circumscribed circle of the triangle 
. Prove that the points 
lie on a circle.This post has been edited 1 time. Last edited by parmenides51, Aug 19, 2021, 10:15 PM
