[/quote]
,
has a right angle at 
.
be the point on line 
such that 
and 
lies between 
.
is chosen so that 
and 
is the bisector of 
.
is chosen so that 
and 
is the bisector of 
.
be the midpoint of 
be the point such that 
is a parallelogram. Prove that 
and 
are concurrent.
satisfying the following conditions:
for all 
Define the set ![\[S = \left\{ \left( \sum_{i=1}^{10} a_i x_i, \sum_{i=1}^{10} a_i y_i \right) : a_1, a_2, \cdots, a_{10} \in \{0, 1\} \right\},\]](http://latex.artofproblemsolving.com/d/e/b/deb50830b9fe10c86ea169c044c1b26f515b2514.png)
then 
，
satisfying: 
is such that 
for any positive integers 
.
such that 
for all positive integers 
.
.
,
,
,
,
in some order.
be an acute triangle with orthocenter 
.
be the point such that the quadrilateral 
is a parallelogram. Let 
be the point on the line 
such that 
.
at 
.
.
and 
be points on circle 
,
,
tangent to 
be any point on arc 
,
,
,
to circles 
.
,
,
,
be points on circle 
equal parts. Construct 
tangent to 
,
,
to circles 
of 
(where 
)
and 
with sum 
.![\[\sum_{i=1}^n \min\{{x_i^2, x_{i+1}}\}\leq \frac{1}{2}.\]](http://latex.artofproblemsolving.com/7/f/0/7f01a8d065c8e03233fe2254946522a5762f80e2.png)
where 
.
of the form 
with 
,
Determine all ordered pairs of polynomials 
with integer coefficients such that 
.
be the reflections of 
,
be the reflections of 
over 
be the circumcenter of triangle 
,
be the orthocenter of triangle 
.
.
is parallel to 
.
,
,
be positive real numbers. Prove the inequality![\[ \frac{1}{a\left(b+1\right)}+\frac{1}{b\left(c+1\right)}+\frac{1}{c\left(a+1\right)}\geq \frac{3}{1+abc}. \]](http://latex.artofproblemsolving.com/d/a/5/da5096e4c732244baf08effc43b5d086444e2c62.png)
.
is a root of the cubic equation 
(real or complex), then 
satisfying the equation for all real numbers 
:![\[
f(x^2 + y + f(y)) = f(x)^2
\]](http://latex.artofproblemsolving.com/0/6/f/06f2c3b16aecf900a28c033c45993564afd80055.png)
to job 
and has a better ability in job 
Y by
Let 
be a triangle with circumcircle 
and circumcenter 
. Let 
and 
represent the 
-excircle and -excenter, respectively. Denote by 
the circle tangent to 
, 
, and on the arc not containing , and similarly for 
. Let the tangency points of 
with line be 
, respectively. Let 
be the intersection point of 
and . Define 
as the point on segment 
such that 
. Suppose that 
intersects again at 
. Let 
be the touch point of the -mixtilinear incircle and , and let 
be the antipode of with respect to . Let 
be the intersection of 
and 
.
Show that the line
is the radical axis of and .
be a triangle with circumcircle 
and circumcenter 
. Let 
and 
represent the 
-excircle and -excenter, respectively. Denote by 
the circle tangent to 
, 
, and on the arc not containing , and similarly for 
. Let the tangency points of 
with line be 
, respectively. Let 
be the intersection point of 
and . Define 
as the point on segment 
such that 
. Suppose that 
intersects again at 
. Let 
be the touch point of the -mixtilinear incircle and , and let 
be the antipode of with respect to . Let 
be the intersection of 
and 
.Show that the line
is the radical axis of and .
