[/quote]
,
be a parallelogram of center 
.
,
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 
.
lies on 
such that 
.
passing trough 
at 
.
.
of positive integers such that
,
.
and 
Prove that
with circumcenter 
,
be an arbitrary point on the circumcircle of triangle 
and that line 
is not parallel to line 
.
be the point on the circumcircle of triangle 
is perpendicular to 
be the point on line 
.
be the points on the circumcircle of triangle 
is a diameter, and 
and 
are parallel. Let 
be the midpoint of 
.
and 
are perpendicular.
and 
are perpendicular.
be nonnegative real numbers such that 
.
are natural numbers satisfy that: 
and 
are prime numbers. Prove that 
is a perfect squared number
Prove that
Y by ImSh95
10. Let 
be an odd number and let 
be an integer such that 
. IN a sports tournament, 
players take part in a series of contests. In each contest, 
players participate, and the scores obtained by the players are the numbers
in some order. Each possible subset of players takes part together in exactly one contest. let the final score of player 
be 
, for each 
. Define 
to be the smallest difference between the final scores of two players, i.e.,
Determine, with proof, the maximum possible value of .
be an odd number and let 
be an integer such that 
. IN a sports tournament, 
players take part in a series of contests. In each contest, 
players participate, and the scores obtained by the players are the numbers
in some order. Each possible subset of players takes part together in exactly one contest. let the final score of player 
be 
, for each 
. Define 
to be the smallest difference between the final scores of two players, i.e.,
Determine, with proof, the maximum possible value of .
