[/quote]
,
and incenter 
and let 
be the midpoint of 
.
,
,
are selected on sides 
,
such that 
,
,
.
intersects 
other than 
.
and 
meet on 
and 
Prove that
,
,
,
,
,
initially contains one coin. The following operations are allowed
,
,
;
,
,
and 
.
coins.
Prove that
its circumcenter, and 
its circumradius. Let 
,
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 
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 
,
.
Y by
In triangle 
, we doubled the median 
and obtained point 
. Points 
are the feet of the altitudes drawn from the vertices 
, respectively, 
is the orthocenter of triangle . Prove that lines 
and 
are perpendicular.
(A.Karlyuchenko)
, we doubled the median 
and obtained point 
. Points 
are the feet of the altitudes drawn from the vertices 
, respectively, 
is the orthocenter of triangle . Prove that lines 
and 
are perpendicular.(A.Karlyuchenko)
This post has been edited 1 time. Last edited by parmenides51, Jul 1, 2021, 5:51 PM
