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 side lengths of a triangle. Prove that![\[\frac{a^3}{(a+b)^3}+\frac{b^3}{(b+c)^3}+\frac{c^3}{(c+a)^3} \leqslant \frac{9}{8}.\]](http://latex.artofproblemsolving.com/5/1/5/515cbd3cd69fee483312f90d4cb29d98ca761f41.png)
and 
. Prove that
Let 
be reals such that 
. Prove that
spheres centered at 
, with radius of 
and , which is remarked as 
and 
, respectively. Given 
points 
outside of (not including the surface of ).
as the number of sets 
such that the midpoint of 
lies entirely inside of .
satisfying 
Prove that 
When does equality hold?
be three positive reals such that 
. Prove that![\[ \frac { x^5-x^2 }{x^5+y^2+z^2} + \frac {y^5-y^2}{x^2+y^5+z^2} + \frac {z^5-z^2}{x^2+y^2+z^5} \geq 0 . \]](http://latex.artofproblemsolving.com/a/1/4/a14adf0f1e35df57c734c7df701d5a67da22dc3c.png)
then
where 
Also, 
and 
is largest real root of the equation
be real numbers such that : 
. Prove that : 
such that the following inequality holds for all 
:![\[
ab + bc + ca \leq \frac{(a + b + c)^2}{3} + k \cdot \max \{ (a - b)^2, (b - c)^2, (c - a)^2 \} \leq a^2 + b^2 + c^2
\]](http://latex.artofproblemsolving.com/1/c/e/1ce66163068b499383adb4a4ecec86c5f4edaec3.png)
be positive real numbers such that : 
. Prove that : 
be positive real numbers such that : . Prove that : 
and its diameter 
are given. Find the locus of the centers of circles inscribed in the triangles having one vertex on and two other vertices on 
(except maybe for the actual circle; that can be reached only when the two points on it coincide). To see this: fix a point 
on and a line 
through , and make the two points 
on move so that is the angle bisector of 
. Let 
. When 
, we get the incenters 
respectively, so, by continuity, the entire segment cut by on belongs to the locus.. This happens iff 
are collinear, so maybe we should disregard , but it can be obtained as the incetner of another triangle constructed in a similar fashion by fixing a point 
on , and taking 
.
?
and 
Prove that 
Prove that
such that 
.
then 
Equality holds iff 
and any cyclic permutations.
