AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
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., 
, 
, and 
. In which cases does equality hold?
be a circumscribed quadrilateral with incircle 
and no two opposite angles equal. Let 
be an arbitrary point on the diagonal 
, which is inside . The segments 
and 
intersect at 
and 
. The tangents to at and intersect at 
. Prove that lies on the line .
be an acute triangle with 
, midpoint 
of side 
, altitude 
(
), and orthocenter 
. A circle passes through points 
and 
, is tangent to line 
, and intersects the circumcircle of triangle at a second point 
. The circumcircle of triangle 
intersects line 
at a second point 
. Prove that the lines 
and 
are perpendicular.
be a convex quadrilateral such that 
. Suppose that a point 
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 
, is 
, is 
, is 
, is 
such that for 
grid, no matter how you paint 
unit squares black, there exists 
square with all unit square painted black.
be a triangle with 
, 
, 
. Let 
, and 
be points lying on sides 
, respectively. Given that 
, and 
are cyclic respectively, and the circumcircle of 
are tangent to line 
, find the length of segment 
.
, we say that a positive integer 
is 
tastic if there exists a number 
such that 
is short, and such that 
is not short for any 
. Let 
be the set of tastic numbers. Consider for 
What is the maximum number of elements in ?
are positive real numbers satisfied 
. Prove that:
satisfy,
Where {
} is a permutation of . Prove that 
and 
into three parts and construct an equilateral triangle from all received parts.
and into three parts and construct an equilateral triangle from all received parts.
be real numbers. Prove that
and 
.
