[/quote]
,
be an acute triangle and let 
be an interior point for it such 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 
Circles 
and 
are internally tangent to 
and 
respectively, and both are tangent to 
Similarly, circles 
and 
are externally tangent to 
and 
respectively.
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.
have the same incenter 
.
is concyclic iff 
is concurrent
be the nonnegative integers in 
where 
.
.
.
such that 
where 
,
?
where 
,
be a scalene triangle with circumcircle 
and circumcircle 
.
and 
as the midpoints of 
and 
,
to 
.
meets 
,
intersects 
.
at 
and 
at 
.
intersect 
,
intersect 
.
is the reflection 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 
and 
Prove that
Y by
Given a semicircle of diameter 
, with 
, be 
a variable string, but of fixed length 
. Let 
be the intersection point of lines 
and 
, and let 
be the intersection point of lines 
and 
.
a) Prove that the lines
and are perpendicular.
b) Determine the locus of the point.
c) Prove that has a constant measure, and determine it based on and 
.
, with 
, be 
a variable string, but of fixed length 
. Let 
be the intersection point of lines 
and 
, and let 
be the intersection point of lines 
and 
.a) Prove that the lines
and are perpendicular.b) Determine the locus of the point
.c) Prove that
has a constant measure, and determine it based on and 
.
.
,
and then 
.
,
.
.
,
and 
.
.
.
,
.
.
.
is the circle 
.
.
