[/quote]
,
to 
row and 
column get symbolized by square 
. Subset 
of squares on the board is called "good" subset if two random squares 
belong to 
with 
or 
are not belong to 
has a right angle at 
.
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 
.
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 
divided into three equal parts. Construct three equal circles 
,
,
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)
Y by
Let 
be a positive integer. There are circles drawn on a chalkboard such that any two circles intersect at 
distinct points and no 
circles pass through the same point. Turbo the snail slides along the circles in the following manner, leaving snail goo behind. Initially he moves on one of the circles in clockwise direction. He keeps sliding along until he reaches an intersection with another circle. Then, he continues his journey on this new circle and also changes the direction he is moving in. We define a snail orbit to be the covering of the whole surface of a circle with turbo's goo, and specifically only a single layer of it. Prove that for every odd there exists at least one configuration of circles with a single snail orbit, and find all such that there is exactly one of the aforementioned configuration type.
be a positive integer. There are circles drawn on a chalkboard such that any two circles intersect at 
distinct points and no 
circles pass through the same point. Turbo the snail slides along the circles in the following manner, leaving snail goo behind. Initially he moves on one of the circles in clockwise direction. He keeps sliding along until he reaches an intersection with another circle. Then, he continues his journey on this new circle and also changes the direction he is moving in. We define a snail orbit to be the covering of the whole surface of a circle with turbo's goo, and specifically only a single layer of it. Prove that for every odd there exists at least one configuration of circles with a single snail orbit, and find all such that there is exactly one of the aforementioned configuration type.
