[/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 
meet at 
.
satisfying both of the following properties:
.
.
and 
for each nonnegative integer 
.
is the greatest integer less than or equal to 
,
Y by
Fix a nonnegative integer 
to define a sequence of integers 
by letting 
be the smallest integer (strictly) greater than 
making 
into a perfect square. Let 
be the set of positive integers not expressible as the difference of two terms of the sequence 
Prove that 
is finite and determine its size in terms of 
to define a sequence of integers 
by letting 
be the smallest integer (strictly) greater than 
making 
into a perfect square. Let 
be the set of positive integers not expressible as the difference of two terms of the sequence 
Prove that 
is finite and determine its size in terms of 
This post has been edited 2 times. Last edited by oVlad, May 14, 2023, 10:18 AM
and proving that 
by using simple bounding. Once you have done this, the number 
will become important and you will get that all differences of the form 
and 
are achievable. This essentially tells you that all large enough odd numbers and all large enough even numbers are expressible; thereby solving the problem. (for the actual construction one needs some more rigour but it's not that hard to recreate; one can prove these are the smallest possible differences and nothing below these can be constructed)
