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.
be a polynomial with integer coefficients such that 
and 
for some integer 
. Prove that there exists no integer 
such that 
.
for which the following statement holds: there exists at least one triple 
of integers such that
and all triples of real numbers, satisfying the equations, are such that 
are integers.
Prove that
Where 
Prove that
Let 
Prove that
Let 
Prove that
, Where 
vertices is given. To each vertex, a natural number from the set 
is assigned. Prove that there are 
vertices 
such that if the numbers 
are assigned to them respectively, then 
and 
is a parallelogram.
be a triangle inscribed in a circle 
with orthocenter 
and altitude 
. Let 
be the midpoint of 
. Line 
meets again at 
. Line 
meets again at 
. Let 
be the orthogonal projection of 
on the line 
. Line 
meets again at 
. Prove that 
.
satisfy 
. Prove that
Also, find the values of for which the equality happens.
such that![\[
x+f(xy)=f(1+f(y))x
\]](http://latex.artofproblemsolving.com/4/3/3/433f95836dad1224754d480aff8432ef033a37cb.png)
for all 
.
be six positive real numbers. Prove that
Prove that
Let 
Prove that
and 
. Prove that
Where 
Where 
