AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
, 
.Find the largest 
that satisfies:
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 in terms of 
be a prime and 
a nonempty set. Suppose that for each element 
there exist elements 
, 
such that 
is divisible by . Prove that 
, where 
denotes the cardinality of 
.
different point-like cities were founded one by one; from the second onwards, each newly founded city and the nearest existing city (the older one, if there were two) were declared sister cities. The surviving map of the empire shows the cities and the distances between them, but not the order in which they were founded. Historians have tried to deduce from the map that each city had at most 41 sister cities.
, give a map from which this deduction can be made.
, this conclusion cannot be drawn from any map.
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 . Prove that is also a cyclic quadrilateral.
and a positive integer , we denote the -fold composition of by 
. A polynomial with real coefficients is called perfect if for each integer there is a positive integer so that 
is an integer. Is it true that for each perfect polynomial , there exists a positive 
so that for each integer there is 
for which is an integer?
meet 
at 
then from Seva teorem
and 
be the feet of altitudes from 
, respectively.Let 
be feet of perpendiculars from 
to the side 
and to the altitude 
and let 
and 
be the feet of perpendiculars from 
and 
to the side , and foot of altitude to 
, respectively.
, and 
with ratio 
.
and 
is Nagel point of the triangle 
and by Nagel line theorem 
are collinear and 
, on the other hand 
are collinear ( Euler line ) and 
,so the triangle 
and the triangle 
similar, so 
parallel to 
and 
.
such that:
,
,
,
,
.
are the incentre, circumcentre and orthocentre of 
respectively. Point 
,
and 
,
.
and 
be 
and 
.
,
,
.
is centroid of the trianle