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 real numbers such that 
. Show that
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?
for which there exists an even natural number 
such that the number![\[
(a - 1)(a^2 - 1)\cdots(a^n - 1)
\]](http://latex.artofproblemsolving.com/9/c/a/9caf4eeb82ff46b5ba55ab4b6bc28f0cace586ec.png)
is a perfect square.
-tuple 
, 
, of symbols from 
, we define a sequence 
according to the following rule: If 
, then 
, where 
if 
(taking 
) and 
is the symbol other than 
if 
. Find all positive integers 
for which there exists some integer 
such that 
.
, such that 
, there exists an integer satisfying 
?
the sum of all positive divisors of .
be non-negative real variables with 
Prove that
Equality holds iff 
or 
where 
, 
, 
is the point on 
such that 
. Let 
be the circle through 
tangent to 
at , and 
the circle through tangent to 
at 
. Let 
be the second intersection of and . Prove that 
lies on 
.
, points and lie on sides and respectively, such that 
, 
, 
, 
and intersects 
at point 
. If 
, then find the lengths of 
and 
.
and 
.
and 
Prove that
Prove that
Prove that
Prove that
and 
.
.
gives 
and then 
.
and 
.