[/quote]
,
such that 
is a perfect square.
such that, for every positive integer 
the integer
is a multiple of 
(Note that 
denotes the greatest integer less than or equal to 
For example, 
and 
)
If the value of 
can be written as 
where m and n are co-prime positive integers then find the value of 
If we set 
then it's the unique polynomial such that for all integers 
In general, for each positive integer k,there is a unique polynomial 
such that 
Find the value of 
for 
and 
if 
then find the coefficient of 
in 
such that:
be a prime and 
a nonempty set. Suppose that for each element 
there exist elements 
,
such that 
is divisible by 
,
denotes the cardinality of 
.
,
,
is divided into cyclic quadrilaterals with pairwise disjoint interiors. None of the vertices of the cyclic quadrilaterals in the decomposition is an interior point of a side of any cyclic quadrilateral in the decomposition or of a side of the quadrilateral 
and a positive integer 
.
is an integer. Is it true that for each perfect polynomial 
so that for each integer 
for which 
with the property that 
is divisible by 
.
Y by sami1618
In triangle 
, let 
be the 
-excenter. Points 
and 
are placed on line 
such that 
is between and 
, and is between and . Moreover, and are the contact points of with the -excircle of triangles 
and 
, respectively. Let 
be the -excenter of triangle 
, and let 
be the reflection of the orthocenter of triangle with respect to its circumcenter. Prove that , , and are collinear.
Proposed by Ali Nazarboland
, let 
be the 
-excenter. Points 
and 
are placed on line 
such that 
is between and 
, and is between and . Moreover, and are the contact points of with the -excircle of triangles 
and 
, respectively. Let 
be the -excenter of triangle 
, and let 
be the reflection of the orthocenter of triangle with respect to its circumcenter. Prove that , , and are collinear.Proposed by Ali Nazarboland
This post has been edited 2 times. Last edited by jannatiar, Mar 4, 2025, 7:58 AM
be the ellipse passing through 
with foci 
and 
and let 
be the ellipse passing through 
be a point the lies on both 
and 
.
be the circle centered at 
and define 
and 
similarly. Then, 
Assume that 
.
is externally tangent to 
that is internally tangent to 
lies on 
,
and 
similarly. Then notice 
,
,
.
.
lies on 
and 
are tangent to 
gives that 
lies on 
lie on 
and 
are tangent to 
,
,
and hence the conjugate 
.
intersects circle 
at 
which implies the existence of a circle tangent at 
using the Bogdanov's theorem as well. The fact that 
was just a fact I heard somewhere and it also appeared in Sharygin 2023 first round. I didn't prove it myself but there exist a lot of different methods to prove that like coordinate bash or even taking the configuration into the 3D space. The fact that 
could be easily proven by 2 point-DIT (having 
)
