[/quote]
,
such that 
, where 
are non-negative reals such that 
.
.
a tangencial quadrilateral that is not a rhombus, let 
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.
positive integers. Product of any one of them with sum of remaining numbers increased by 
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 
is a prime.(Don't get mad at me,I've used the search function and did not see a correct and complete solution anywhere.)
![\[
2^a + 3^b + 1 = 6^c
\]](http://latex.artofproblemsolving.com/1/9/e/19eb5efa2ccdf54cd277339f8d3b99f75168779d.png)
is inscribed in circle 
.
and 
and 
be the midpoints of sides 
and 
,
and 
meet 
and 
,
and 
.
satisfying both of the following properties:
.
.
and 
for each nonnegative integer 
.
is the greatest integer less than or equal to 
,
Y by
In the accompanying figure, semicircles with centers
and 
have radii 
and 
, respectively. Furthermore, they are internally tangent to the circle of diameter 
. Also the semicircles with centers and 
are externally tangent to each other. The circle with center 
is internally tangent to the semicircle with diameter and externally tangent to the others two semicircles. Determine the value of the radius of the circle with center .
and 
have radii 
and 
, respectively. Furthermore, they are internally tangent to the circle of diameter 
. Also the semicircles with centers and 
are externally tangent to each other. The circle with center 
is internally tangent to the semicircle with diameter and externally tangent to the others two semicircles. Determine the value of the radius of the circle with center .This post has been edited 1 time. Last edited by parmenides51, Oct 2, 2021, 1:41 PM
Therefore 
.
