[/quote]
,
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 
of positive integers such that
,
.
and 
Prove that
Y by
Let 
and 
be positive integers. Kellem and Carmen play the following game: initially, the number 
is on the board. Starting with Kellem and alternating turns, they add powers of to the previous number on the board, such that the new value on the board does not exceed . The player who writes wins. Determine, for each pair 
, who has the winning strategy.
Note: A power of is a number of the form 
, where 
is a non-negative integer.
and 
be positive integers. Kellem and Carmen play the following game: initially, the number 
is on the board. Starting with Kellem and alternating turns, they add powers of to the previous number on the board, such that the new value on the board does not exceed . The player who writes wins. Determine, for each pair 
, who has the winning strategy.Note: A power of
is a number of the form 
, where 
is a non-negative integer.
