[/quote]
,
Prove that
are integers whose greatest common divisor is 1. Let 
be a set of integers with the following properties:
,
.
(not necessarily distinct), 
.
,
,
.
.
,
such that 
is divisible by 
,
.
![\[
\frac{1}{3}|AB^3 - AD^3| \le |BC^3 - CD^3| \le 3|AB^3 - AD^3|.
\]](http://latex.artofproblemsolving.com/5/5/e/55ee0fba8392ee81b7b37da45d99755ea6d6754a.png)
such that 
for all 
with parallel sides 
and 
,
on line 
outside segment 
inside segment 
such that 
.
the point of intersection of 
,
the point of intersection of 
be the midpoint of segment 
if and only if 
.
with an area of 
,
.
,
.
.
.
.
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)
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)
Y by Adventure10
A lattice point 
in the plane is said to be visible from the origin 
if the line segment 
does not contain any other lattice points. Show that for any positive integer 
, there is square 
of area 
such that none of the lattice points inside the square is visible from the origin.
in the plane is said to be visible from the origin 
if the line segment 
does not contain any other lattice points. Show that for any positive integer 
, there is square 
of area 
such that none of the lattice points inside the square is visible from the origin.
to be different primes we can set 
?
then a point is invisible. Then, we know by CRT there exists a solution with primes 
s.t. if 
is the coordinate of the bottom left corner, we have 
:
.
co-ordinates inorder for it to be invisible.
denote the left-bottom most point of the square.
for 
such that: 
.
and 
such that
for all 
.
works.
