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 the number of positive integers that divide the integer 
. For all positive integral divisors 
of 
, what is the sum of numbers 
?
such that the value of the
is a square of an integer.
, let 
be all positive integers smaller than that are coprime to . Find all 
such that 
for all 
is the largest positive integer that divides both 
and 
. Integers and are coprime if 
.
, , 
, 
, where and 
are odd primes and![\[2^ap^b=(p+2)^c-1.\]](http://latex.artofproblemsolving.com/5/8/5/58518a8ebed14836b72af743ff100eae2efba5e2.png)
be a triangle with 
let 
be the circumcircle of 
and let 
be its radius. Point 
is chosen on 
such taht 
and point 
is the foot of the perpendicular from to 
. Ray 
mets again at 
. Point 
is chosen on line 
such that 
and 
lie on a line in that order. Finally, let 
be a point satisfying 
and 
. Prove that lies on ., so that the equation 
has two real roots, so that the one is the square of the other.
such that: ![\[ x = \sqrt{ x - \frac{1}{x} } + \sqrt{ 1 - \frac{1}{x} } \]](http://latex.artofproblemsolving.com/e/c/f/ecf7fb3c2805290f2cb07b7fdd7c73e1c7a55b99.png)
be a convex pentagon such that 
. Suppose that the midpoint of 
is the circumcenter of triangle 
. Let 
be the circumcenter of triangle 
.
passes through the midpoint of segment 
.
be a cyclic pentagon inscribed in a circle of centre which has angles 
. Show that the diagonals 
and 
meet at a point belonging to the diameter .
and 
are the incenter and the 
-excenter of triangle , respectively.
be the midpoint of arc 
on the circumcircle, and be the foot of the to 
. The line 
intersects the circumcircle again at 
. For
on the circumcircle of triangle , let 
intersect at 
. Prove
are concyclic.
. 
are the midpoints of 
respectively. is the midpoint of arc , is symmetric to through . Let 
be the incenter and tangent of angles of triangle 
respectively. Prove that 
have a common point
be real numbers such that 
P
![\[
x^4 + 4y^4 + 2z^4 + 2z^4 \geq 8 \lvert xyz^2 \rvert \geq 8xyz^2.
\]](http://latex.artofproblemsolving.com/b/e/4/be46e5cb76cf132ae849db753c77057fe8035b8a.png)
![\[
3x^4 + 8xyz^2 \leq 3x^4 + x^4 + 4y^4 + 2z^4 + 2z^4 = 20.
\]](http://latex.artofproblemsolving.com/5/9/b/59b60f986e58bbcaa7dfb95234052695cae3c2a5.png)