[/quote]
,
be real numbers. Prove that
be real numbers such that ![\[ \frac{1}{x^2-1} + \frac{1}{y^2-1} + \frac{1}{z^2-1} = 1. \]](http://latex.artofproblemsolving.com/8/f/5/8f5d49c9b44e3db3cd85fea7b6521e1dc02d4269.png)
Prove that ![\[ \frac{1}{x+1} + \frac{1}{y+1} + \frac{1}{z+1} \le 1. \]](http://latex.artofproblemsolving.com/c/0/4/c0438d6eba686d0a11fd4decb4afa9a14aa50bac.png)
be real numbers. Prove that
where the indices are taken modulo 8, i.e., 
,
,
.
be a circumscribed quadrilateral with incircle 
and no two opposite angles equal. Let 
,
intersect 
.
be an acute triangle with 
,
of side 
,
)
.
and 
,
,
.
intersects line 
at a second point 
.
and 
are perpendicular.
.
on the side 
satisfies the equality![\[AE\cdot ED + BE^2=CD\cdot AE.\]](http://latex.artofproblemsolving.com/d/7/5/d75cc3a449a56b43eac574cc44e9932a2e63b33b.png)
.
then number of positive integral divisors of 
,
,
,
,
such that for 
grid, no matter how you paint 
unit squares black, there exists 
square with all unit square painted black.
,
,
.
,
be points lying on sides 
,
,
are cyclic respectively, and the circumcircle of 
are tangent to line 
,
.
,
is 
tastic if there exists a number 
such that 
is short, and such that 
is not short for any 
.
be the set of 
What is the maximum number of elements in 
are positive real numbers satisfied 
.
satisfy,
Where 
}
and 
.
Y by
On the hypotenuse 
of the right-angled triangle 
, a point 
is chosen such that 
. Let 
be a point on the perpendicular line from point to the line 
, equidistant from the points and 
. Also let 
denote the midpoint of the segment . Prove that line 
is tangent to the circumcircle of the triangle 
.
of the right-angled triangle 
, a point 
is chosen such that 
. Let 
be a point on the perpendicular line from point to the line 
, equidistant from the points and 
. Also let 
denote the midpoint of the segment . Prove that line 
is tangent to the circumcircle of the triangle 
.
be the circumcenter of 
.
.
,
.
is cyclic, we have 
.
.
is the center of spiral similarity sending 
to 
,
.
,
