AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
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 is increasing.
Determine all the local minima of (if any) and all the local maxima of (if any) 
, angle 
. Point 
lies on 
such that 
. Line parallel to 
passing trough intersects 
at 
. Prove that 
.
of positive integers such that
+1 w
, then it must not contain edge 
.
with circumcenter 
and orthocenter 
, let 
be an arbitrary point on the circumcircle of triangle such that does not lie on line 
and that line 
is not parallel to line 
. Let 
be the point on the circumcircle of triangle such that 
is perpendicular to , and let 
be the point on line such that 
. Let and 
be the points on the circumcircle of triangle such that 
is a diameter, and 
and 
are parallel. Let 
be the midpoint of 
.
and 
are perpendicular.
and 
are perpendicular.
be nonnegative real numbers such that 
. Find min
are natural numbers satisfy that: 
and 
are prime numbers. Prove that 
is a perfect squared number
denote the sum of the sum of the factors of all positive 
, for example, 
is the sum of the factors of 
added to the sum of the factors of 
and so on until the sum of the factors of 
, which would be 
. Let 
denote 
. Show that 
or find a counterexample
checkered board is cut into several squares with integer side length. The size of a square is it's side length.
checkered board is cut into several squares with integer side length. The size of a square is it's side length.
so maximum possible amount 
is also impossible
and 
as big squares square because not shared area is maximum big squares - contradiction
we can build next example:
, then cut 
into 
and 
in first rectangle 
and 
in second, 
and 
in third, 
and 
into and 
and 
into 
square because not shared area is maximum 
square because not shared area is maximum 
it is required to have at least 
square
and 
Prove that
Prove that
squares of distinct sizes.![[asy]
size(7cm);
import geometry;
for(int i = 0; i<21; ++i){
draw((i,0)--(i,20), linewidth(0.2));
draw((0,i)--(20,i), linewidth(0.2));
}
draw((0,20)--(0,10)--(10,10)--(10,20)--cycle, blue+linewidth(1));
draw((10,20)--(10,11)--(19,11)--(19,20)--cycle, blue+linewidth(1));
draw((0,10)--(0,2)--(8,2)--(8,10)--cycle, blue+linewidth(1));
draw((8,10)--(8,5)--(13,5)--(13,10)--cycle, blue+linewidth(1));
draw((13,11)--(13,4)--(20,4)--(20,11)--cycle, blue+linewidth(1));
draw((0,0)--(2,0)--(2,2)--(0,2)--cycle, blue+linewidth(1));
draw((16, 0)--(20,0)--(20,4)--(16,4)--cycle, blue+linewidth(1));
draw((13,1)--(16,1)--(16,4)--(13,4)--cycle, blue+linewidth(1));
draw((10,2)--(11,2)--(11,3)--(10,3)--cycle, blue+linewidth(1));
[/asy]](http://latex.artofproblemsolving.com/6/f/c/6fca4b28cb893c0f274d808ad2954ebe961efe98.png)