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 an acute triangle and let 
be an interior point for it such that 
. Prove that
When does equality hold?
be positive reals. Prove that there is a cyclic permutation 
of 
such that the inequality:![\[
\frac{a}{xa + yb + zc} + \frac{b}{xb + yc + za} + \frac{c}{xc + ya + zb} \geq \frac{3}{x + y + z}
\]](http://latex.artofproblemsolving.com/5/5/3/553527fc40a1b7d29dbdfb4b5e63635308f46c85.png)
and 
.
be real numbers such that 
.![\[
3 + (2 - \sqrt{3}) \cdot \frac{(b-c)^2}{b+(\sqrt{3}-1)c} \leq a+b+c
\]](http://latex.artofproblemsolving.com/5/a/8/5a83eaa7470bcca6546599ff4e296e4efec69b16.png)
and determine all the cases when the equality occurs.
have the same incenter 
.Prove that 
is concyclic iff 
is concurrent
be the nonnegative integers in 
where 
. Define 
. Prove that 
.
such that 
where 
, and the statement is still true when 
? such that 
where 
, and the statement is still true when ?.
be a scalene triangle with circumcircle 
and circumcircle 
. Denote 
and 
as the midpoints of 
and 
, respectively. The altitude from 
to intersects again at 
. The line 
meets a second time at 
, and 
intersects at 
. The perpendicular bisector of intersects 
at 
and 
at 
. Let the circumcircle of 
intersect again at 
, and let the line 
intersect at 
.
is the reflection of over , prove that the intersection of 
and 
lies on the circumcircle of 
.
be real numbers such that 
and 
and determine the equality cases.
+1 w
be positive real integer such that 
Prove that
such that 
and 
satisfying :![\[n^{2p}=m^2+n^2+p+1\]](http://latex.artofproblemsolving.com/7/4/e/74ee592cb8b3f3a3996ea34f40b78fde9d3672fc.png)
and 
are integers and 
is a prime number.
Approach: I'm trying to factor 
to get 
.
such that 
divides 
.
such that 
.
.
intersects three of the lines 
,
,
,
,
