[/quote]
,
with 
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 
.
.
.
,
and 
such that 
.
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 
of positive integers such that
,
.
and 
Prove that
Y by Adventure10, Mango247
A boy lives in a small island in which there are three roads at every junction. He starts
from his home and walks along the roads. At each junction he would choose to turn
to the road on his right or left alternatively, i.e., his choices would be . . ., left, right,
left,... Prove that he will eventually return to his home.
from his home and walks along the roads. At each junction he would choose to turn
to the road on his right or left alternatively, i.e., his choices would be . . ., left, right,
left,... Prove that he will eventually return to his home.
be the graph of this island(each junction is a vertex and each road is an edge).Let 
then 
.
with 
as following:
,
.
,
uniquely.
possible values
and 2 possibility for 
)
such that 
so obviously 
and we are done.
