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.
and 
with integer coefficients is called \emph{important} if from the divisibility of both differences 
and 
by 
, it follows that both 
and 
are divisible by 100. Does there exist such an important pair of polynomials 
, 
, such that the pair 
and 
is also important?
is given. A cube with side length 
is made up of 
unit cubes, each of which is either black or white. It turns out that among any 
cubes that share a common vertex and form a 
cube, there are at most 
black cubes. What is the maximum number of black cubes that could have been used?
such that there exists a pair 
of positive integers, such that 
is not divisible by the cube of any prime, and 
be positive real numbers with 
. Prove that
be reals such that 
. Prove that
Let be reals such that 
. Prove that
and 
, such that for any positive integer 
, there is a subset 
of 
with cardinality 
, and for any 
with 
, the difference 
is not a perfect square.
, the sides 
and 
are equal. Let 
be the touch point the inscribed circle of the given pentagon with side 
, and 
is the center of this circle. Prove that points 
are collinear.
, the sides and are equal. Let be the touch point the inscribed circle of the given pentagon with side , and is the center of this circle. Prove that points are collinear.
with an inscribed circle; the points of tangency are 
of the sides 
.
, then also 
.
,
.
, so 
are collinear.
are positive real numbers, 
then
![\[
\frac{(x+y+z)^5 - x^5 - y^5 - z^5}{(x+y)(y+z)(z+x)} = 0
\]](http://latex.artofproblemsolving.com/7/8/f/78f8e6c2570684631dc288144e13568d71e68ec8.png)
and 
Prove that
