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 nonzero real numbers such that 
. Assume that 
Show that for any odd integer 
, 
be the unit circle in the complex plane. Let 
be the map given by 
. We define 
and 
for 
. The smallest positive integer 
such that 
is called the period of 
. Determine the total number of points in 
of period 
.
)
and let 
be positive integers such that 
. Prove that 
and determine when equality holds.
denote the set of natural numbers, and let 
, 
, be nine distinct tuples in 
. Show that there are three distinct elements in the set 
whose product is a perfect cube.
satisfy the equality, 
prove that the triangle must have a right angle.
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
, let 
be the midpoint of 
. Let 
be the foot of the perpendicular from 
to 
. Suppose that the circumcircle of triangle 
intersects line 
at two distinct points 
and 
. Let 
be the midpoint of 
. Prove that 
.
be six positive real numbers. Prove that
and 
. Prove that
Where 
