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.
, then prove that the following inequality holds:![\[
\sum_{\text{cyc}} \sqrt{1 + \left(x\sqrt{1 + y^2} + y\sqrt{1 + x^2}\right)^2} \geq \sum_{\text{cyc}} xy + 2\sum_{\text{cyc}} x
\]](http://latex.artofproblemsolving.com/9/5/5/955863a0cf8747ae45b736b0631f243d3908eb84.png)
such that has exactly 2000 prime divisors and divides 
?
and 
is a subset of 
such that if 
then 
Find the smallest of 
that satisfy the following property: if 
, 
, 
, 
are all the positive divisors of 
with 
, then 
divides 
for every 
.
, in which 
: 
, but the proof is important.
+1 w
and 
Prove that
satisfying ![$$(ab+bc+ca)\left[\frac{1}{(a+b)^{2}}+\frac{1}{(b+c)^{2}}+\frac{1}{(c+a)^{2}}\right]\ge \frac{9}{4}+k\cdot\frac{a(a-b)(a-c)+b(b-a)(b-c)+c(c-a)(c-b)}{(a+b+c)^{3}}$$](http://latex.artofproblemsolving.com/f/6/e/f6ed10f7fff1cc94edd8f451e75718a0916a8bfa.png)
holds for all 
be non-negative real numbers such that 
Prove that
be a scalene triangle with 
, and let 
be the foot of the altitude from 
. Let 
be a point in the interior of the segment 
. Let 
be the point on the segment 
such that 
. Similarly, let 
be the point on the segment 
such that 
. The circumcircle of triangle 
intersects segment 
at a second point 
(other than ). Prove that 
.
, whether or not the polynomial 
irreducible over 
?
such that the equation in the unknown 
:
admits 
roots: 
s.t. 
.
is a subgroup of 
with 4 elements, hence 
for all 
and 
. In particular ![$Z[i]\subset A$](http://latex.artofproblemsolving.com/5/0/d/50d33e14432872b85f66d025ea971c0f0ccf1444.png)
. We have 
for some 
and 
, hence 
. Also 
hence 
for some 
and 
is an algebraic integer. Hence ![$\alpha\in Z[i]$](http://latex.artofproblemsolving.com/9/f/5/9f557018ba023b02c9c6d92543be87f061995d30.png)
and we are done.
![$\underset{n\to \infty }{\mathop{\lim }}\,\sqrt[n]{lcm(1,2,3,...,n)}=e$](http://latex.artofproblemsolving.com/a/1/3/a139e5cb990f3386de109c701f8edc71b4054712.png)
be such that the set 
is a ring with respect to the usual operations of 
If the ring 
has exactly four invertible elements, prove that ![$A= \mathbb{Z}[i].$](http://latex.artofproblemsolving.com/6/b/c/6bc4a63fbd716536c436cad654841755835342d7.png)
