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.
with the property that ![\[f(x-f(y))=f(f(x))-f(y)-1\]](http://latex.artofproblemsolving.com/f/2/5/f25cc1e8ae0be1fd02b347fd94be4fab88af1d46.png)
holds for all 
.
be a function such that
for all 
. Find all such functions 
.
be a function such that
for all 
. Determine all such functions .
. Find all 
such that for all 
be the set of real numbers. Find all functions 
where there exist a real constant 
such that 
holds for all reals 
, 
, 
that satisfy 
.
with its center 
. There are two fixed points 
and one moving point 
on . The midpoint of the line segment 
is 
. 
is a fixed point on . Line 
intersects
at 
, and line 
intersects 
at 
. such that is always tangent to 
when varies.
be a non constant polynomial with real coefficients. Prove that the system of simultaneous equations —
has finitely many solutions .
such that
be a scalene triangle with orthocenter 
and circumcenter . Denote by , 
the midpoints of 
, 
. Suppose the circle 
with diameter meets the circumcircle of at 
, and meets line 
at a point 
. The tangent to at 
meets line 
at 
. Show that the circumcircles of 
and 
intersect at a point 
on 
.
Why we can't write 
as 
such that
for all 
) for all their time and support. We received immense help from TestX01 for the proof of claim 2.19 and it's associated lemma.
be a Chebyshev polynomial of the second kind. If n>2 and x > 2 is a integer, Could 
be a prime number?
be a Chebyshev polynomial of the second kind. If n>2 and x > 2 is a integer, Could be a prime number?
even 
is even .
odd we can check that 
for 
and 
is divisible with the minimal polynomial of 
and 
.