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.
are integers whose greatest common divisor is 1. Let 
be a set of integers with the following properties:
, 
.
(not necessarily distinct), 
.
, if 
, then 
. must be equal to the set of all integers.
.
, there exist indices 
such that 
is divisible by .
, find the minimum possible value of 
.
be a quadrilateral circumscribed about a circle, whose interior and exterior angles are at least 60 degrees. Prove that![\[
\frac{1}{3}|AB^3 - AD^3| \le |BC^3 - CD^3| \le 3|AB^3 - AD^3|.
\]](http://latex.artofproblemsolving.com/5/5/e/55ee0fba8392ee81b7b37da45d99755ea6d6754a.png)
such that 
for all 
with parallel sides 
and 
, assume that there exist points 
on line 
outside segment , and 
inside segment 
such that 
. Denote by 
the point of intersection of and 
, and by 
the point of intersection of and . Let 
be the midpoint of segment , assume it does not lie on line . Prove that belongs to the circumcircle of 
if and only if belongs to the circumcircle of 
.
with an area of 
, with 
. If 
, Hence find the value of 
.
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)
points inside a circle of radius 
Show that there are at least 
of these points that are inside a circle of radius 
the area of a circle radius 
pigeons
must have at least 
points inside
Prove that
.
.
.
such that 
.![$$2 \sqrt[4]{\frac{a}{b+c-a}}\ge 2 +\frac{2a^2-b^2-c^2}{(a+b)(a+c)}.$$](http://latex.artofproblemsolving.com/f/0/7/f0731fbe5c83aedc8474975f930f7194c08cc070.png)
