[/quote]
,
and 
be 2 distinct positive integers.
be in an integer. Prove why ?
are non-negative reals such that 
.
.
be lengths of tangents from 
to the incircle of the quadrilateral which center is 
.
be the midpoints of 
resp. Prove that![\[ \frac{MI}{IN}=\frac{a+c}{b+d} \]](http://latex.artofproblemsolving.com/e/7/0/e709acb8b17c0466a385ba8e7a89bb1bf16c72af.png)
is given. It turns out that for every positive integer number 
,
and 
are positive, they have the same number of divisors. Is it necessarily true that 
?
with 
be the intersection of 
and 
and define 
in a similar way. Furthermore, let 
.
,
are concyclic.
is divisible with sum of all 
![$P \in \mathbb R[x,y]$](http://latex.artofproblemsolving.com/8/a/0/8a0576299e976c51607959fb68c7c742b2a49c6a.png)
with ![$P \not\in \mathbb R[x] \cup \mathbb R[y]$](http://latex.artofproblemsolving.com/a/f/7/af71ba8c643d5bd8a2928fa6714ccd91713bd641.png)
and 
homeomorphismes of 
,
is an homoemorphisme too.
,
.
be the second intersection point of the circumcircles of the triangles 
and 
.
,
and 
are the circumcenters of the triangles 
are concyclic.
where 
![\[
2^a + 3^b + 1 = 6^c
\]](http://latex.artofproblemsolving.com/1/9/e/19eb5efa2ccdf54cd277339f8d3b99f75168779d.png)
with maximums 
and 
.
,
can be expressed as 
where 
and 
.
Let 
.
Let 
.
Let 
.
Y by HamstPan38825, Adventure10, Mango247
Let 
be an isosceles trapezoid with 
, 
, and 
. Denote by 
and 
the incircles of 
and 
respectively. Suppose that both and have radii of length 
. The square of the distance between the centers of the two circles can be written in the form 
, where 
, 
, and 
are positive integers such that is not divisible by the square of any prime. What is 
?
be an isosceles trapezoid with 
, 
, and 
. Denote by 
and 
the incircles of 
and 
respectively. Suppose that both and have radii of length 
. The square of the distance between the centers of the two circles can be written in the form 
, where 
, 
, and 
are positive integers such that is not divisible by the square of any prime. What is 
?
as the incenters of 
and 
and 
.
.
.
from 
,
.
.
