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 
.
that satisfy the equation 
.
be a circle, and let a line 
intersect at two points, and 
Circles 
and 
are internally tangent to at points 
and 
respectively, and both are tangent to at a common point 
Similarly, circles 
and 
are externally tangent to at and respectively, and are tangent to at points 
and 
respectively. is 
the segment 
has a length of 
and 
find the length of segment 
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.
+1 w
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.
be positive real integer such that 
Prove that
such that 
and 
, there is a circle of radius 
, and a small semicircle, touching each other in pairs, like shown in the figure. Find the radius of the small semicircle.
, there is a circle of radius , and a small semicircle, touching each other in pairs, like shown in the figure. Find the radius of the small semicircle.
and 
Prove that
Draw the line connecting the center of the circle with radius 
and a hypotenuse of 
Using the Pythagorean Theorem gives 
Expanding gives 
Bringing everything with 
on one side and moving everything else to the other side gives 
Thus, the radius of the small semicircle is 