[/quote]
,
and 
be positive real numbers such that 
.![\[
\frac{x^{3}}{(1 + y)(1 + z)}+\frac{y^{3}}{(1 + z)(1 + x)}+\frac{z^{3}}{(1 + x)(1 + y)}
\geq \frac{3}{4}.
\]](http://latex.artofproblemsolving.com/5/6/5/56568d27a296f1c900fabbb1042549cd2e552e29.png)
,![\[ \frac{a}{\sqrt{a^2 + 8bc}} + \frac{b}{\sqrt{b^2 + 8ca}} + \frac{c}{\sqrt{c^2 + 8ab}} \geq 1. \]](http://latex.artofproblemsolving.com/b/2/f/b2ff456632d4dd3b9ed93e4cba437bdd34d54cd8.png)
,
,
cut three common chords concurrent at 
.
,
which are internally tangent to all 
.
be a parallelogram of center 
.
,
and 
such that 
.
be given . On the extension of the segment 
beyond 
,
is chosen arbitrarily. Let the line through 
intersect the line 
at the point 
.
be the intersection of the lines 
and 
.
is tangent to the circumcircle of the triangle 
.
be any point. Let 
be projection point from 
.
cuts circumcircle 
at 
.
be antipode of 
are cyclic.
,
be real numbers, such that 
.
.
.
such that 
.
passing trough 
at 
.
.
Compute -![\begin{align*}
\frac{\mathrm{d}}{\mathrm{d}x} \bigg[ \int_{0}^{e^x} \log ( t ) \cos^4 ( t ) \mathrm{d}t \bigg]
\end{align*}](http://latex.artofproblemsolving.com/8/6/f/86fb48db976012106bd30f74cb6ada8c6e6d7800.png)
For 
define 
Determine the open interval(s) (if any) where 
is decreasing and all the open interval(s) (if any) where 
Determine all the local minima of 
Y by Adventure10, Mango247
Let be two natural numbers 
and 
pairwise disjoint sets of natural numbers 
each having 
elements, such that no element of 
is divisible by an element of 
for any natural number 
Determine the number of ordered pairs
such that 
and such that 
for any 
Radu Bumbăcea
and 
pairwise disjoint sets of natural numbers 
each having 
elements, such that no element of 
is divisible by an element of 
for any natural number 
Determine the number of ordered pairs
such that 
and such that 
for any 
Radu Bumbăcea
divide everything in 
,
Then just pick a huge prime 
consist of the numbers 
As long as 
is not in any of the 
vertices labeled 
with 
edges, there exists an 
so that there exists a path from 
(Here indices are modulo 
When 
,
When 
,![$f: [m] \rightarrow [m]$](http://latex.artofproblemsolving.com/b/e/c/bec2221f615ef143634d0375965b2699056d007d.png)
as follows. For each 
,
,
is the smallest positive integer so that 
does not point to 
It's easy to see that if 
,
(Again, indices modulo 
so that 
Suppose there did exist such 
If 
,
to see that either there is a path from 
for some 
,
to 
In the former case, the induction is complete. In the latter case, as 
)
by the definition of 
,
to 
If 
,
are all distinct numbers. Then, we know that 
and 
If 
,
,
we are also done. Therefore, there can be no edge between 
and 
Now, if 
,
and we're done. Otherwise, if 
,
and we're done. Therefore, assume there is no edge between 
Let there be 
cool integers in 
Call a non-edge a pair of integers 
where 
non-edges consisting of consecutive positive integers. We wish to show that there are at least 
,
There are 
integers 
where 
Then, there are actually only 
of these non-edges, because 
of them are counted twice. Let these 
From the previous paragraph, we know that for each 
,
is also a non-edge. These non-edges are clearly distinct from each other. They are also distinct from the 
cool non-edges because 
is not cool 
aren't consecutive. Hence, putting all of these together gives us 
non-edges. Hence, there must be at most 
non-edges, a clear contradiction. The induction is complete, and therefore the lemma is proven.
edges, and so the probabilistic method now easily yields that there are at most 
edges overall.
