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.
is called perfect if the sum of its positive divisors 
is twice , that is, 
. For example, 
is a perfect number since the sum of its positive divisors is 
, which is twice . Prove that if is a positive perfect integer, then:![\[
\sum_{p|n} \frac{1}{p + 1} < \ln 2 < \sum_{p|n} \frac{1}{p - 1}
\]](http://latex.artofproblemsolving.com/0/c/0/0c08f527e87f7aae4ca5226f19bd084249551fe3.png)
where the sums are taken over all prime divisors 
of .
for which 
is a perfect square.
having the following property: for any integers 
whose sum is not divisible by , there exists an index 
such that none of the numbers 
is divisible by . Here, we let 
when 
.
has 
on 
-angle bisector such that 
. A line passes perpendicular to 
cuts 
at 
, respectively. 
is the intersection of 
and 
. 
is defined similarly. 
and 
cuts 
at 
, respectively. Prove that: 
.
be an arbitrary point on the side 
in a triangle 
. Let 
and 
be the intersection of the lines parallel to 
and 
through with and . Let 
be the intersection of 
with 
. Let 
be the midpoint of and 
the intersection of 
with . Let 
be the intersection of 
with . Prove that 
, 
and 
are concurrent at 
.
and are collinear. There are surprisingly many properties in the configuration posted earlier :P
so for any integer , where 
, it is possible to find three natural numbers 
(
) such that 
.
, for which exist natural numbers, such that 
and 
.
with the following property: the 
positive divisors of have a permutation 
such that for 
, the number 
is a perfect square.
and 
.
is a root of 
, try to prove or disprove that :
of triangle 
touches 
at points 
, respectively. Let be the intersection point of 
and 
. Points and 
lie on 
and 
, respectively, such that 
and 
. Let 
is the intersection of 
and 
. Prove that 
.
is a complex matrix with 
prove that is self-adjoint, i.e., that 
over all triples 
of distinct real numbers such that
,
,
,
.
and its absolute and relative error, known that its absolute error is equal or lower than 
