be a non-constant monic polynomial with integer coefficients and let 
be an infinite sequence. Prove that there are infinitely many primes, each of which divides at least one term of the sequence 
.
positive real numbers such that 
. Prove that :
be an integer, and 
can divide 
let 
be all distinct prime divisors of 
is an integer. ( where 
is defined as the number of positive integers 
that are relatively prime to 
be real numbers . Prove that
Where 
Y by baopbc, doxuanlong15052000, Tawan, Adventure10, Mango247
Let 
be an arbitrary triangle and 
is the circumcenter of 
.Points 
lie on 
,respectively such that the reflection of 
WRT 
is tangent to circumcircle of 
.Prove that the circumcircle of triangle 
is tangent to circumcircle of triangle 
.
be an arbitrary triangle and 
is the circumcenter of 
.Points 
lie on 
,respectively such that the reflection of 
WRT 
is tangent to circumcircle of 
.Prove that the circumcircle of triangle 
is tangent to circumcircle of triangle 
.
Let 
be the symmetric of 
through 
.
be the intersection of 
and 
.
through 
which gives 
are concyclic.
:
and 
are tangent at 
if and if only:
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over 
touches 
at 
be the Miquel point of 
w.r.t. 
.
,
,
by simmetry, therefore: 
as claimed. 
hence, 
and 
are tangent to each other at 
By the Miquel 
concur at some point 
Hence 
tangent at 
in 
and 
are concyclic, we are just directly done by inversion at 
swapping 
and 
,
In that case, 
So this motivates us to define 
