AoPS Academy
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)
such that, for every positive integer 
the integer
is a multiple of 
(Note that 
denotes the greatest integer less than or equal to 
For example, 
and 
)
.
, there exist indices 
such that 
is divisible by .
, find the minimum possible value of 
.
. The point 
is in the interior of . The following ratio equalities hold:![\[\angle PAD:\angle PBA:\angle DPA=1:2:3=\angle CBP:\angle BAP:\angle BPC\]](http://latex.artofproblemsolving.com/9/3/8/9388a91325842f1ec67a4693f0f10af46ed66c6d.png)
Prove that the following three lines meet in a point: the internal bisectors of angles 
and 
and the perpendicular bisector of segment 
.
for which there exists an integer 
such that 
is a divisor of 
.
with 
. The circumcircle 
of has radius 
. There is a point in the interior of the line segment such that 
and the length of 
is . The perpendicular bisector of 
intersects at the points 
and 
. is the incentre of triangle 
.
and 
Find the value of 
be the similicenter of 
; 
touch point of a tangent to 
through ; perpendicular to 
wrt 
is 
;
, the radical axis of , cuts 
at ; a point of intersection of 
and ;
cuts at 
; let the bisector of
cuts at 
passing through is tangent to .
be the similicenter of ; touch point of a tangent to through ; perpendicular to wrt is ;, the radical axis of , cuts at ; a point of intersection of and ; cuts at ; let the bisector of cuts at passing through is tangent to .
the centers of 
and their radii be 
.The point on 
with satisfy 
is the required center of circle. This is done by marking intersection of hyperbola (having foci and constant 
) with line .It is easy to find synthetic sol for this.
and 
Prove that![$$3\left(\sqrt[3] 2+\frac{1}{\sqrt[3] 2} -1\right) \geq a+b+c\geq \frac{3+\sqrt{11}}{2}$$](http://latex.artofproblemsolving.com/f/6/6/f66042b4d07212741245953a8940280c94a52988.png)
![$$\frac{3}{2}\left(4+\sqrt[3] 4-\sqrt[3] 2\right) \geq a+b+c+ab+bc+ca\geq \frac{3(1+\sqrt{11})}{2}$$](http://latex.artofproblemsolving.com/0/5/6/056db8bf4d80242d705c265b6a64e063c8fca15a.png)
Prove that
given circles and 
?